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Welcome back to Story Problem Office Hours.
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In this week's main episode, a line scratched onto a turning drum became a number that could compare earthquakes.
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Charles Richter and Bino Gutenberg used a logarithmic scale to express that wide range of short practical set of numbers.
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Today we're going to go through the mathematical concept more in depth, and hopefully in a way that will be understandable, because honestly, this is a bit of a confusing concept.
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What is a logarithm actually asking?
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What does a difference of two mean on a logarithmic scale?
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And what happens when the unknown number appears in an exponent?
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These concepts can be confusing, and teaching it to your kids or trying to brush up on these concepts can be quite challenging.
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In this episode, let's build the toolkit first.
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Then we'll use it on two challenge problems at the end of the study hall sheet.
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Those are problems nine and ten, and we're going to take our time with both of them.
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But here's the sentence I want in your head for the entire episode.
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A logarithm asks, what exponent produced this number?
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And here is the four-part habit we'll use.
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Ask, translate, compare, solve.
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I'm Behirbani.
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This is Story Problem Office Hours.
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Logarithms, ask the exponent.
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Complete the logarithmic statement.
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10 to the fourth power equals 10,000.
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So the base 10 logarithm of 10,000 equals blank.
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Okay, so what do I know?
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I know the base is 10.
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I know the result is 10,000, and I can see the exponent that connects them.
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10 to the fourth power equals 10,000.
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What is the logarithm asking me to name?
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It's asking for that exponent.
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The exponent is four.
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So 10 to the fourth equals 10,000 means log base 10 of 10,000 equals 4.
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The answer is 4.
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Notice that nothing new happened to the three numbers.
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The base stayed 10, the result stayed 10,000, the exponent four moved from a raised position in the exponential statement to the answer in the logarithmic statement.
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A logarithm is simply asking you to find an exponent.
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The base 10 logarithm of 10,000 means 10 raised to what power equals 10,000?
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The answer is 4 because 10 to the fourth power is 10,000.
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So let's write the relationship in a general form.
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Don't worry about memorizing the letters.
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Focus on what each one represents, the base, the exponent, and the result.
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b to the y equals x means log base b of x equals y.
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There are always three roles.
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The base is the number being raised to a power.
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The exponent tells us which power, and the result is what that power produces.
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In 10 to the fourth equals 10,000, 10 is the base, 4 is the exponent, and 10,000 is the result.
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In the logarithmic form, the base is still 10 and the result is still 10,000.
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The logarithm asks us for the exponent, which is four.
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Here is a base two example, the base two logarithm of thirty-two equals five.
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Translate it by asking the question in ordinary language.
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Two raised to what power gives thirty-two?
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Two to the fifth power gives thirty-two.
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So in other words, log base two of thirty-two equals five means two to the fifth power equals thirty-two.
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Here's another one.
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The base five logarithm of x equals three.
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Translate this before calculating.
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To reword this or to translate it, log base five of x equals three, which is the same as five to the third equals x, and x in this case is one hundred twenty-five.
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That's the important move.
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Keep the base, identify the exponent, and identify the result.
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Then move between the two forms.
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So before touching a calculator, name the three roles base, exponent, and result.
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A logarithmic statement and an exponential statement describe the same relationship from opposite directions.
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So let's talk about the toolkit.
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Ask what the problem wants.
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Is it asking for an exponent, a missing number, a ratio between two measurements, or time hidden inside an exponential model?
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The word logarithm does not tell you the whole job.
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The question does.
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And translate before calculating.
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Rewrite a logarithm as an exponent whenever that makes the relationship easier to see.
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Rewrite an exponential equation with a logarithm when the unknown appears in the exponent.
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Translation is not extra work.
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It's how you expose the part that you actually really need.
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And remember what negative exponents mean.
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Evaluate the base 10 logarithm of 0.001.
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Ask the exponent question 10 to what power gives 1000?
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10 to the negative third power.
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So 0.001 equals 10 to the negative third, log base 10 of 0.001 is negative 3.
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The negative answer does not mean the input is negative.
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The input is positive, 0.001.
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The negative exponent tells us that the number is between 0 and 1.
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10 to the negative third means 1 divided by 10 to the third, or 1 divided by 1000.
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On a logarithmic scale, subtract the steps and then find the factor.
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Suppose earthquake A has a magnitude 7.2 and earthquake B has magnitude 5.2.
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The printed numbers are two units apart.
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7.2 minus 5.2 equals 2.
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That subtraction tells us the number of powers of 10 steps between them.
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2 steps means 10 squared.
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10 squared is 100.
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In the simplified original local magnitude comparison, earthquake A has 100 times the measured wave amplitude of earthquake B.
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Amplitude ratio equals 10 squared equals 100.
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The common mistake is to say two times larger because the magnitudes differ by 2, but the scale is not counting ordinary equal additions underneath.
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Each step represents multiplication by 10.
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Subtract to find the steps, then use the difference as an exponent.
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Direction matters too.
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If the question asks how many times larger A is than B, place the larger amplitude over the smaller amplitude.
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Reversing the comparison gives one hundredth.
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That answers how large B is relative to A, which is a different question.
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Read the direction of the particular scale as well.
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That's the next step.
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The pH scale uses a negative logarithm.
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That negative sign reverses the direction.
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A lower pH means a greater hydrogen ion concentration.
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Compare pH with pH 5.
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The values are two units apart, so the concentration factor is still 10 squared or 100, but the lower number, pH, has the greater hydrogen ion concentration.
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It has 100 times the hydrogen ion concentration of the pH 5 solution.
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So don't assume that a higher number always means more of what is being measured.
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On the earthquake magnitude scale, a higher number means a larger measured wave amplitude.
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The pH scale works in the opposite direction.
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A lower pH means a higher hydrogen ion concentration.
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So first find how many steps apart the two values are, and then check which direction that particular scale runs.
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In this problem set, we do use a couple of log rules.
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We haven't gone over these, but these are relatively simple rules to know and to understand.
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So use the log rules as translations of multiplication and division.
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So the two logarithm rules will matter in these challenge problems.
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And those two rules are log of a times b in parentheses equals log of a plus log of b.
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And log of a divided by b in parentheses equals log of a minus log of b.
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Here is why those rules work.
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Let me just go through that to make that a little bit more simplified.
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Let's use powers of 10.
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100 is 10 to the second power, and 1's thousand is 10 to the third power.
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If we multiply them, then we get 100,000, which is 10 to the 5th power.
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The new exponent 5 comes from adding the original exponents, 2 plus 3.
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A logarithm simply tells us those exponents.
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The base 10 logarithm of 100 is 2, the base 10 logarithm of 1000 is 3.
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When we multiply 100 by 1000, their logarithms add 2 plus 3 equals 5.
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Division works the same way in reverse.
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1000 divided by 100 is 10.
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The exponents subtract.
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3 minus 2 equals 1, which is why multiplying numbers becomes addition when we use logarithms and dividing numbers become becomes subtraction.
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Let's take problem number 8.
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For that problem, we're asking for f of x equals log base 2 of x, which expression is equal to f of eight x minus f of x?
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In this case, if f of x is the base 2 logarithm of x, then f of eight x minus f of x becomes the base two logarithm of 8x minus the base two logarithm of x.
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The quotient rule tells us it allows us to combine the difference into one logarithm.
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The base two logarithm of 8x divided by x.
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And you can see here the variable cancels out.
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So for a positive x, the x terms cancel.
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We are left with the base two logarithm of eight.
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2 to the third power is 8, so the answer is 3.
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The log base 2 of 8 is 3.
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The meaning is more important than the symbol manipulation.
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Multiplying any positive input by 8 moves its base 2 logarithm upward by 3 because 8 itself is 2 cubed.
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Let's go through challenge problem number 9.
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Time hidden in an exponent.
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A culture begins with 600 cells and grows according to P of T equals 600 times 1.25 raised to the T power, where T is measured in hours.
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When will the culture reach 2000 cells?
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Round to the nearest tenth of an hour.
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So what do I know?
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The culture starts with 600 cells.
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Each hour it's multiplied by 1.25.
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The target is 2000 cells.
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I'm trying to find time and it appears as the exponent.
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So first let's make the model mean something to us.
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Before solving, let's translate the model into ordinary language.
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So for the equation P of T equals 600 times 1.25 raised to the T power, we want to translate that into something that's meaningful for us.
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The 600 is the starting amount.
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At time zero, the exponent is zero.
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1.25 to the zero power is one.
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So p of zero is six hundred times one or six hundred cells.
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The 1.25 is the hourly growth multiplier.
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It means the culture keeps all of its previous cells, which is 100%, and adds 25% more during each hour.
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That's why the multiplier is 125%, written as 1.25.
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After one hour, the culture has 600 times 1.25 or 750 cells.
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After two hours, we multiply by 1.25 again.
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The repeated multiplication is what places time in the exponent.
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So let's put the target into the equation.
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We want to know when the population reaches 2000.
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So replace P of T with 2000.
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600 times 1.25 raised to the t power equals 2000.
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Our goal is to get the expression containing t by itself.
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Right now it's multiplied by 600.
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Divide both sides by 600.
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So if you do that, 1.25 raised to the t power equals 2000 divided by 600, and that equals 10 over 3, which in decimal form is about 3.33.
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This line says that the culture must grow to about 3.33 times its starting size.
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The question is now very clean.
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1.25 raised to what power gives us about 3.33.
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You could give yourself an idea of the answer before using any logarithms.
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A quick estimate helps us understand the answer and catch any future mistakes by using a calculator.
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After five hours, the model gives us about 1,831 cells.
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After six hours, it gives us about 2,289 cells.
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So P of 5 is about 1,831, and P of 6 is about 2,289.
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2,000 falls between those two values.
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So the answer must be between 5 and 6 hours.
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If our calculator doesn't give us a value in that range, then we know something went wrong.
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So here's the power of a logarithm.
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Use it to bring the exponent down.
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There's no ordinary step like addition, subtraction, multiplication, or division that changes t from an exponent into a factor we can isolate.
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This is exactly when a logarithm is useful.
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Take the logarithm of both sides.
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You may use the common logarithm button labeled log or the natural logarithm button labeled ln.
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Either one works as long as you use the same kind of logarithm on both sides.
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Log 1.25 raised to the t power equals log 10 over 3 or 3.33.
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Now use the power rule.
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An exponent inside a logarithm can come down in front.
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t times log 1.25 equals log 10 over 3.
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This is the key change.
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A moment ago, t appeared as the exponent.
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Now it's an ordinary factor multiplied by the logarithm of 1.25.
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Divide both sides by that logarithm and t equals log 10 over 3 divided by log 1.25.
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When we use our calculator, that gives us about 5.396 hours rounded to the nearest tenth.
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That's about 5.4 hours.
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So t is about 5.4 hours.
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So put 5.4 back into the model, 600 times 1.25 raised to the 5.4 power is about 2,002 cells.
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The tiny difference comes from rounding.
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That's exactly where we want it to land.
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And our answer agrees with the estimate.
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5.4 lies between 5 and 6 hours.
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When the unknown is the exponent, isolate the exponential part first.
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Then take logarithms, use the power rule to bring the exponent down and divide.
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The calculation identifies the exponent precisely.
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1.25 raised to what power produces the required growth factor.
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Now let's try challenge problem number 10.
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And this one's kind of neat because I think it it incorporates all the lessons that we've learned so far today.
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The question is solve the base two logarithm of x minus 1 plus the base 2 logarithm of x minus 3 equals 3.
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And the clue is to be sure to check the domain of both logarithms.
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So what do I know?
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Both logarithms use base 2.
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They are being added, so the product rule may combine them, but before I do any algebra, I need to ask which x values are even allowed.
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So let's set the boundaries.
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A real logarithm can only take a positive input, not zero, not a negative number, positive only.
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Why?
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Because a positive base such as two stays positive when we raise it to any real power.
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Two raised to some real exponent can produce a large positive number or a small positive fraction, but it cannot produce zero or a negative number.
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So for x minus one, that has to be greater than zero, so x has to be one or greater, and x minus three has to be greater than zero, so that gives x greater than three.
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X must be greater than three.
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So both conditions must be true at the same time.
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The stricter condition is x greater than three.
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That's our domain.
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Any answer at or below three is not allowed in the original equation.
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So our domain is x must be greater than three.
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Writing the domain now gives us a boundary to check.
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The algebra may produce more than one candidate later.
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We'll compare each candidate with the original domain to see which ones are valid.
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So now let's combine those two logarithms.
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The logarithms have the same base and they are added.
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The product rule says that a sum of logarithms becomes the logarithm of a product.
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A small number example shows why.
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The base two logarithm of four is two and the base two logarithm of two is one.
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Add them and you get three.
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Multiply the inputs four times two and you get eight.
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The base two logarithm of eight is also three.
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So the base two logarithm of four plus the log base two of two equals two plus one, which is three.
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So the log base two of eight.
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Now apply that rule to the expression in the problem.
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Log base two of, in parentheses, x minus one times x minus three equals three.
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We've not multiplied the expressions out yet.
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We've simply replaced two logarithms with one logarithm of their product.
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Now we translate the raw logarithm into exponential form.
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So let's return to that first master question.
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The base two logarithm of this product equals three.
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That means two raised to the third power equals the product.
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So x minus one times x minus three equals two raised to the third power, which is eight.
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There it is.
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The logarithms are gone.
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What remains is an algebra equation.
00:20:25.200 --> 00:20:27.359
So then we solve the algebra.
00:20:27.599 --> 00:20:30.400
Multiply two expressions on the left.
00:20:30.720 --> 00:20:33.599
x times x is x squared.
00:20:33.759 --> 00:20:41.599
The two middle terms are negative three x and negative one x, which combines to negative four x.
00:20:41.920 --> 00:20:45.200
And negative one times negative three is positive three.
00:20:45.359 --> 00:20:49.519
So x squared minus four x plus three equals eight.
00:20:50.319 --> 00:20:54.000
Subtract eight from both sides so that one side equals zero.
00:20:54.319 --> 00:20:57.920
x squared minus four x minus five equals zero.
00:20:58.160 --> 00:20:59.440
And now we factor.
00:20:59.519 --> 00:21:01.599
This is a quadratic equation.
00:21:01.759 --> 00:21:06.240
We need two numbers that multiply to negative five and add to negative four.
00:21:06.400 --> 00:21:09.519
Those numbers are negative five and positive one.
00:21:10.400 --> 00:21:14.400
A product equals zero when at least one factor equals zero.
00:21:14.480 --> 00:21:22.799
So x minus five equals zero giving x equals five, or x plus one equals zero, giving x equals negative one.
00:21:23.200 --> 00:21:27.680
So our candidates are x equals five or x equals negative one.
00:21:28.160 --> 00:21:29.759
Now let's return to the domain.
00:21:29.920 --> 00:21:32.319
This is where the word candidates matter.
00:21:32.480 --> 00:21:39.599
The quadratic equation produced two values, but our original equation contained logarithms.
00:21:40.000 --> 00:21:42.960
We need to check both values in the original problem.
00:21:43.119 --> 00:21:49.200
Start with x equals five, then x minus one is four, and x minus three is two.
00:21:49.599 --> 00:21:53.839
Both inputs are positive, so both logarithms exist.
00:21:54.079 --> 00:22:00.960
The base two logarithms of four is two, the base two logarithm of two is one, so two plus one equals three.
00:22:01.119 --> 00:22:04.000
So x equals five works.
00:22:04.400 --> 00:22:06.880
Now x equals negative one.
00:22:07.200 --> 00:22:08.400
Let's try that.
00:22:09.039 --> 00:22:13.119
Then x minus one is negative two, and x minus three is negative four.
00:22:13.200 --> 00:22:16.480
In the real number system, those logarithms are not defined.
00:22:16.720 --> 00:22:24.880
Negative one may solve the quadratic we created after combining and translating, but it does not solve the original logarithmic equation.
00:22:25.279 --> 00:22:30.880
x equals negative one, the logarithm inputs are negative, so that's not allowed.
00:22:31.119 --> 00:22:33.839
So the only solution is x equals five.
00:22:33.920 --> 00:22:36.000
So that's our final answer.
00:22:37.119 --> 00:22:41.119
Algebra can produce a number that's not permitted by the original problem.
00:22:41.200 --> 00:22:43.920
With logarithms, check the inputs.
00:22:44.160 --> 00:22:47.359
Every logarithm must receive a positive number.
00:22:47.599 --> 00:22:52.480
Solve the algebra, then check every result in the original equation.
00:22:59.200 --> 00:23:01.759
So that is the logarithms toolkit.
00:23:01.920 --> 00:23:06.880
Start with one question: what exponent produced this number?
00:23:07.039 --> 00:23:27.279
Then use the four-part habit, ask what the problem wants, translate between logarithmic and exponential form, compare values on a logarithmic scale by finding the difference and turning that difference into a factor, solve and then check that answer is allowed in the original problem.
00:23:27.519 --> 00:23:29.599
And remember the two challenge moves.
00:23:29.839 --> 00:23:38.400
If time is hidden in an exponent, isolate the exponential expression and use a logarithm to bring the exponent down.
00:23:38.799 --> 00:23:45.200
If logarithms are added or subtracted, use the product or quotient rule when the bases match.
00:23:45.440 --> 00:23:50.400
And whenever a variable sits inside a logarithm, track the domain.
00:23:50.720 --> 00:23:54.160
You do not need to hold every rule in your head at once.
00:23:54.319 --> 00:23:56.079
Keep the structure in view.
00:23:56.400 --> 00:23:58.240
Base, exponent, result.
00:23:58.960 --> 00:24:03.440
Multiplication underneath becomes addition on the logarithmic scale.
00:24:03.759 --> 00:24:06.319
Division underneath becomes subtraction.
00:24:06.720 --> 00:24:12.720
The equal steps on the printed scale can represent very large ratios in the real world.
00:24:13.119 --> 00:24:18.799
All ten problems are there in the study hall section at studyproblem.org.
00:24:19.119 --> 00:24:26.319
Work through the earlier problems first and then return to problems 9 and 10 and see whether you can explain every step out loud.
00:24:27.200 --> 00:24:34.880
If you can explain why the step is okay and what it means, you understand way more than just the answer.
00:00:08.240 --> 00:00:11.039
Welcome back to Story Problem Office Hours.
00:00:11.279 --> 00:00:18.000
In this week's main episode, a line scratched onto a turning drum became a number that could compare earthquakes.
00:00:18.239 --> 00:00:26.879
Charles Richter and Bino Gutenberg used a logarithmic scale to express that wide range of short practical set of numbers.
00:00:27.120 --> 00:00:36.159
Today we're going to go through the mathematical concept more in depth, and hopefully in a way that will be understandable, because honestly, this is a bit of a confusing concept.
00:00:46.159 --> 00:00:48.399
What is a logarithm actually asking?
00:00:48.560 --> 00:00:52.399
What does a difference of two mean on a logarithmic scale?
00:00:52.560 --> 00:00:56.240
And what happens when the unknown number appears in an exponent?
00:00:56.479 --> 00:01:03.200
These concepts can be confusing, and teaching it to your kids or trying to brush up on these concepts can be quite challenging.
00:01:03.439 --> 00:01:06.239
In this episode, let's build the toolkit first.
00:01:06.480 --> 00:01:10.799
Then we'll use it on two challenge problems at the end of the study hall sheet.
00:01:11.040 --> 00:01:14.719
Those are problems nine and ten, and we're going to take our time with both of them.
00:01:14.879 --> 00:01:19.359
But here's the sentence I want in your head for the entire episode.
00:01:19.599 --> 00:01:23.920
A logarithm asks, what exponent produced this number?
00:01:24.159 --> 00:01:26.640
And here is the four-part habit we'll use.
00:01:26.879 --> 00:01:29.760
Ask, translate, compare, solve.
00:01:30.000 --> 00:01:31.359
I'm Behirbani.
00:01:31.599 --> 00:01:33.519
This is Story Problem Office Hours.
00:01:33.680 --> 00:01:36.480
Logarithms, ask the exponent.
00:01:44.480 --> 00:01:46.400
Complete the logarithmic statement.
00:01:46.560 --> 00:01:49.599
10 to the fourth power equals 10,000.
00:01:49.840 --> 00:01:54.319
So the base 10 logarithm of 10,000 equals blank.
00:01:56.079 --> 00:01:57.840
Okay, so what do I know?
00:01:58.000 --> 00:01:59.920
I know the base is 10.
00:02:00.159 --> 00:02:04.719
I know the result is 10,000, and I can see the exponent that connects them.
00:02:04.959 --> 00:02:08.000
10 to the fourth power equals 10,000.
00:02:08.400 --> 00:02:10.879
What is the logarithm asking me to name?
00:02:11.039 --> 00:02:12.879
It's asking for that exponent.
00:02:13.120 --> 00:02:14.879
The exponent is four.
00:02:15.120 --> 00:02:22.960
So 10 to the fourth equals 10,000 means log base 10 of 10,000 equals 4.
00:02:23.280 --> 00:02:24.560
The answer is 4.
00:02:24.800 --> 00:02:27.439
Notice that nothing new happened to the three numbers.
00:02:27.680 --> 00:02:38.479
The base stayed 10, the result stayed 10,000, the exponent four moved from a raised position in the exponential statement to the answer in the logarithmic statement.
00:02:38.879 --> 00:02:42.159
A logarithm is simply asking you to find an exponent.
00:02:42.319 --> 00:02:49.039
The base 10 logarithm of 10,000 means 10 raised to what power equals 10,000?
00:02:49.520 --> 00:02:54.000
The answer is 4 because 10 to the fourth power is 10,000.
00:02:54.479 --> 00:02:57.280
So let's write the relationship in a general form.
00:02:57.439 --> 00:02:59.360
Don't worry about memorizing the letters.
00:02:59.599 --> 00:03:04.240
Focus on what each one represents, the base, the exponent, and the result.
00:03:05.120 --> 00:03:11.680
b to the y equals x means log base b of x equals y.
00:03:12.400 --> 00:03:14.400
There are always three roles.
00:03:14.560 --> 00:03:17.360
The base is the number being raised to a power.
00:03:17.599 --> 00:03:22.960
The exponent tells us which power, and the result is what that power produces.
00:03:23.360 --> 00:03:30.479
In 10 to the fourth equals 10,000, 10 is the base, 4 is the exponent, and 10,000 is the result.
00:03:30.879 --> 00:03:36.240
In the logarithmic form, the base is still 10 and the result is still 10,000.
00:03:36.560 --> 00:03:40.479
The logarithm asks us for the exponent, which is four.
00:03:40.719 --> 00:03:46.319
Here is a base two example, the base two logarithm of thirty-two equals five.
00:03:46.639 --> 00:03:50.159
Translate it by asking the question in ordinary language.
00:03:50.319 --> 00:03:52.800
Two raised to what power gives thirty-two?
00:03:53.439 --> 00:03:56.319
Two to the fifth power gives thirty-two.
00:03:56.560 --> 00:04:05.759
So in other words, log base two of thirty-two equals five means two to the fifth power equals thirty-two.
00:04:06.560 --> 00:04:07.680
Here's another one.
00:04:07.919 --> 00:04:12.319
The base five logarithm of x equals three.
00:04:13.120 --> 00:04:15.280
Translate this before calculating.
00:04:26.639 --> 00:04:40.240
To reword this or to translate it, log base five of x equals three, which is the same as five to the third equals x, and x in this case is one hundred twenty-five.
00:04:40.560 --> 00:04:42.160
That's the important move.
00:04:42.319 --> 00:04:46.639
Keep the base, identify the exponent, and identify the result.
00:04:47.279 --> 00:04:49.759
Then move between the two forms.
00:04:50.800 --> 00:04:56.240
So before touching a calculator, name the three roles base, exponent, and result.
00:04:56.560 --> 00:05:03.120
A logarithmic statement and an exponential statement describe the same relationship from opposite directions.
00:05:04.480 --> 00:05:06.879
So let's talk about the toolkit.
00:05:07.120 --> 00:05:09.040
Ask what the problem wants.
00:05:09.279 --> 00:05:17.600
Is it asking for an exponent, a missing number, a ratio between two measurements, or time hidden inside an exponential model?
00:05:17.839 --> 00:05:21.279
The word logarithm does not tell you the whole job.
00:05:21.600 --> 00:05:22.959
The question does.
00:05:23.199 --> 00:05:25.360
And translate before calculating.
00:05:25.600 --> 00:05:30.879
Rewrite a logarithm as an exponent whenever that makes the relationship easier to see.
00:05:31.040 --> 00:05:36.399
Rewrite an exponential equation with a logarithm when the unknown appears in the exponent.
00:05:36.720 --> 00:05:38.560
Translation is not extra work.
00:05:38.639 --> 00:05:41.600
It's how you expose the part that you actually really need.
00:05:41.839 --> 00:05:44.160
And remember what negative exponents mean.
00:05:44.399 --> 00:05:49.199
Evaluate the base 10 logarithm of 0.001.
00:05:49.680 --> 00:05:54.639
Ask the exponent question 10 to what power gives 1000?
00:05:55.439 --> 00:05:57.360
10 to the negative third power.
00:05:57.600 --> 00:06:08.800
So 0.001 equals 10 to the negative third, log base 10 of 0.001 is negative 3.
00:06:09.040 --> 00:06:12.079
The negative answer does not mean the input is negative.
00:06:12.160 --> 00:06:15.600
The input is positive, 0.001.
00:06:15.759 --> 00:06:19.439
The negative exponent tells us that the number is between 0 and 1.
00:06:19.680 --> 00:06:25.759
10 to the negative third means 1 divided by 10 to the third, or 1 divided by 1000.
00:06:26.160 --> 00:06:30.240
On a logarithmic scale, subtract the steps and then find the factor.
00:06:30.480 --> 00:06:37.519
Suppose earthquake A has a magnitude 7.2 and earthquake B has magnitude 5.2.
00:06:37.839 --> 00:06:40.800
The printed numbers are two units apart.
00:06:41.120 --> 00:06:44.879
7.2 minus 5.2 equals 2.
00:06:45.199 --> 00:06:49.680
That subtraction tells us the number of powers of 10 steps between them.
00:06:49.920 --> 00:06:52.160
2 steps means 10 squared.
00:06:52.399 --> 00:06:54.399
10 squared is 100.
00:06:54.560 --> 00:07:03.519
In the simplified original local magnitude comparison, earthquake A has 100 times the measured wave amplitude of earthquake B.
00:07:03.839 --> 00:07:08.160
Amplitude ratio equals 10 squared equals 100.
00:07:08.399 --> 00:07:18.879
The common mistake is to say two times larger because the magnitudes differ by 2, but the scale is not counting ordinary equal additions underneath.
00:07:19.120 --> 00:07:21.839
Each step represents multiplication by 10.
00:07:22.079 --> 00:07:26.639
Subtract to find the steps, then use the difference as an exponent.
00:07:26.959 --> 00:07:28.240
Direction matters too.
00:07:28.399 --> 00:07:35.519
If the question asks how many times larger A is than B, place the larger amplitude over the smaller amplitude.
00:07:35.759 --> 00:07:38.720
Reversing the comparison gives one hundredth.
00:07:38.959 --> 00:07:44.079
That answers how large B is relative to A, which is a different question.
00:07:44.480 --> 00:07:46.720
Read the direction of the particular scale as well.
00:07:46.800 --> 00:07:47.600
That's the next step.
00:07:47.759 --> 00:07:50.720
The pH scale uses a negative logarithm.
00:07:50.879 --> 00:07:53.439
That negative sign reverses the direction.
00:07:53.600 --> 00:07:57.839
A lower pH means a greater hydrogen ion concentration.
00:07:58.079 --> 00:08:00.720
Compare pH with pH 5.
00:08:00.959 --> 00:08:12.800
The values are two units apart, so the concentration factor is still 10 squared or 100, but the lower number, pH, has the greater hydrogen ion concentration.
00:08:13.040 --> 00:08:18.079
It has 100 times the hydrogen ion concentration of the pH 5 solution.
00:08:18.319 --> 00:08:23.360
So don't assume that a higher number always means more of what is being measured.
00:08:23.519 --> 00:08:30.079
On the earthquake magnitude scale, a higher number means a larger measured wave amplitude.
00:08:30.240 --> 00:08:32.639
The pH scale works in the opposite direction.
00:08:32.799 --> 00:08:36.960
A lower pH means a higher hydrogen ion concentration.
00:08:37.200 --> 00:08:43.600
So first find how many steps apart the two values are, and then check which direction that particular scale runs.
00:08:43.759 --> 00:08:47.200
In this problem set, we do use a couple of log rules.
00:08:47.360 --> 00:08:53.279
We haven't gone over these, but these are relatively simple rules to know and to understand.
00:08:53.519 --> 00:08:57.840
So use the log rules as translations of multiplication and division.
00:08:58.080 --> 00:09:02.000
So the two logarithm rules will matter in these challenge problems.
00:09:02.159 --> 00:09:10.320
And those two rules are log of a times b in parentheses equals log of a plus log of b.
00:09:10.639 --> 00:09:18.399
And log of a divided by b in parentheses equals log of a minus log of b.
00:09:18.879 --> 00:09:20.720
Here is why those rules work.
00:09:20.879 --> 00:09:24.320
Let me just go through that to make that a little bit more simplified.
00:09:24.559 --> 00:09:26.399
Let's use powers of 10.
00:09:26.639 --> 00:09:33.039
100 is 10 to the second power, and 1's thousand is 10 to the third power.
00:09:33.279 --> 00:09:38.080
If we multiply them, then we get 100,000, which is 10 to the 5th power.
00:09:38.240 --> 00:09:43.279
The new exponent 5 comes from adding the original exponents, 2 plus 3.
00:09:43.519 --> 00:09:46.799
A logarithm simply tells us those exponents.
00:09:46.960 --> 00:09:52.960
The base 10 logarithm of 100 is 2, the base 10 logarithm of 1000 is 3.
00:09:53.200 --> 00:09:59.519
When we multiply 100 by 1000, their logarithms add 2 plus 3 equals 5.
00:09:59.840 --> 00:10:02.320
Division works the same way in reverse.
00:10:02.559 --> 00:10:05.600
1000 divided by 100 is 10.
00:10:05.840 --> 00:10:07.440
The exponents subtract.
00:10:07.600 --> 00:10:16.799
3 minus 2 equals 1, which is why multiplying numbers becomes addition when we use logarithms and dividing numbers become becomes subtraction.
00:10:18.320 --> 00:10:20.320
Let's take problem number 8.
00:10:20.879 --> 00:10:33.039
For that problem, we're asking for f of x equals log base 2 of x, which expression is equal to f of eight x minus f of x?
00:10:33.519 --> 00:10:49.200
In this case, if f of x is the base 2 logarithm of x, then f of eight x minus f of x becomes the base two logarithm of 8x minus the base two logarithm of x.
00:10:49.600 --> 00:10:56.159
The quotient rule tells us it allows us to combine the difference into one logarithm.
00:10:56.399 --> 00:11:01.039
The base two logarithm of 8x divided by x.
00:11:01.440 --> 00:11:05.200
And you can see here the variable cancels out.
00:11:05.519 --> 00:11:09.440
So for a positive x, the x terms cancel.
00:11:09.679 --> 00:11:13.039
We are left with the base two logarithm of eight.
00:11:13.360 --> 00:11:17.120
2 to the third power is 8, so the answer is 3.
00:11:17.360 --> 00:11:20.399
The log base 2 of 8 is 3.
00:11:20.639 --> 00:11:24.399
The meaning is more important than the symbol manipulation.
00:11:24.639 --> 00:11:34.080
Multiplying any positive input by 8 moves its base 2 logarithm upward by 3 because 8 itself is 2 cubed.
00:11:34.240 --> 00:11:36.720
Let's go through challenge problem number 9.
00:11:37.039 --> 00:11:39.279
Time hidden in an exponent.
00:11:40.080 --> 00:11:52.639
A culture begins with 600 cells and grows according to P of T equals 600 times 1.25 raised to the T power, where T is measured in hours.
00:11:52.879 --> 00:11:55.360
When will the culture reach 2000 cells?
00:11:55.600 --> 00:11:58.080
Round to the nearest tenth of an hour.
00:11:59.120 --> 00:12:00.559
So what do I know?
00:12:00.720 --> 00:12:03.519
The culture starts with 600 cells.
00:12:03.759 --> 00:12:07.440
Each hour it's multiplied by 1.25.
00:12:07.919 --> 00:12:10.720
The target is 2000 cells.
00:12:10.879 --> 00:12:14.960
I'm trying to find time and it appears as the exponent.
00:12:15.519 --> 00:12:19.440
So first let's make the model mean something to us.
00:12:19.679 --> 00:12:24.720
Before solving, let's translate the model into ordinary language.
00:12:25.039 --> 00:12:36.559
So for the equation P of T equals 600 times 1.25 raised to the T power, we want to translate that into something that's meaningful for us.
00:12:36.720 --> 00:12:39.039
The 600 is the starting amount.
00:12:39.200 --> 00:12:42.639
At time zero, the exponent is zero.
00:12:43.279 --> 00:12:46.720
1.25 to the zero power is one.
00:12:47.200 --> 00:12:52.879
So p of zero is six hundred times one or six hundred cells.
00:12:53.200 --> 00:12:57.120
The 1.25 is the hourly growth multiplier.
00:12:57.279 --> 00:13:05.919
It means the culture keeps all of its previous cells, which is 100%, and adds 25% more during each hour.
00:13:06.080 --> 00:13:12.000
That's why the multiplier is 125%, written as 1.25.
00:13:12.720 --> 00:13:18.720
After one hour, the culture has 600 times 1.25 or 750 cells.
00:13:18.960 --> 00:13:23.039
After two hours, we multiply by 1.25 again.
00:13:23.279 --> 00:13:27.840
The repeated multiplication is what places time in the exponent.
00:13:28.159 --> 00:13:30.720
So let's put the target into the equation.
00:13:30.879 --> 00:13:34.960
We want to know when the population reaches 2000.
00:13:35.200 --> 00:13:38.240
So replace P of T with 2000.
00:13:38.480 --> 00:13:44.720
600 times 1.25 raised to the t power equals 2000.
00:13:44.960 --> 00:13:49.840
Our goal is to get the expression containing t by itself.
00:13:50.080 --> 00:13:52.720
Right now it's multiplied by 600.
00:13:53.039 --> 00:13:55.519
Divide both sides by 600.
00:13:55.919 --> 00:14:10.159
So if you do that, 1.25 raised to the t power equals 2000 divided by 600, and that equals 10 over 3, which in decimal form is about 3.33.
00:14:10.639 --> 00:14:16.639
This line says that the culture must grow to about 3.33 times its starting size.
00:14:16.879 --> 00:14:18.799
The question is now very clean.
00:14:19.039 --> 00:14:26.159
1.25 raised to what power gives us about 3.33.
00:14:26.720 --> 00:14:31.039
You could give yourself an idea of the answer before using any logarithms.
00:14:31.360 --> 00:14:38.000
A quick estimate helps us understand the answer and catch any future mistakes by using a calculator.
00:14:38.240 --> 00:14:43.360
After five hours, the model gives us about 1,831 cells.
00:14:43.519 --> 00:14:47.600
After six hours, it gives us about 2,289 cells.
00:14:47.759 --> 00:14:54.799
So P of 5 is about 1,831, and P of 6 is about 2,289.
00:14:55.279 --> 00:14:57.759
2,000 falls between those two values.
00:14:57.840 --> 00:15:00.480
So the answer must be between 5 and 6 hours.
00:15:00.639 --> 00:15:04.960
If our calculator doesn't give us a value in that range, then we know something went wrong.
00:15:05.120 --> 00:15:06.879
So here's the power of a logarithm.
00:15:07.039 --> 00:15:09.360
Use it to bring the exponent down.
00:15:09.600 --> 00:15:18.960
There's no ordinary step like addition, subtraction, multiplication, or division that changes t from an exponent into a factor we can isolate.
00:15:19.279 --> 00:15:21.919
This is exactly when a logarithm is useful.
00:15:22.080 --> 00:15:24.240
Take the logarithm of both sides.
00:15:24.480 --> 00:15:30.879
You may use the common logarithm button labeled log or the natural logarithm button labeled ln.
00:15:31.120 --> 00:15:35.840
Either one works as long as you use the same kind of logarithm on both sides.
00:15:36.240 --> 00:15:43.440
Log 1.25 raised to the t power equals log 10 over 3 or 3.33.
00:15:43.919 --> 00:15:45.279
Now use the power rule.
00:15:45.440 --> 00:15:49.919
An exponent inside a logarithm can come down in front.
00:15:50.320 --> 00:15:56.720
t times log 1.25 equals log 10 over 3.
00:15:56.960 --> 00:15:58.159
This is the key change.
00:15:58.320 --> 00:16:00.720
A moment ago, t appeared as the exponent.
00:16:00.879 --> 00:16:06.559
Now it's an ordinary factor multiplied by the logarithm of 1.25.
00:16:07.120 --> 00:16:15.519
Divide both sides by that logarithm and t equals log 10 over 3 divided by log 1.25.
00:16:16.080 --> 00:16:22.159
When we use our calculator, that gives us about 5.396 hours rounded to the nearest tenth.
00:16:22.320 --> 00:16:24.559
That's about 5.4 hours.
00:16:24.639 --> 00:16:27.120
So t is about 5.4 hours.
00:16:27.759 --> 00:16:38.080
So put 5.4 back into the model, 600 times 1.25 raised to the 5.4 power is about 2,002 cells.
00:16:38.320 --> 00:16:40.240
The tiny difference comes from rounding.
00:16:40.320 --> 00:16:42.399
That's exactly where we want it to land.
00:16:42.639 --> 00:16:44.720
And our answer agrees with the estimate.
00:16:44.879 --> 00:16:48.240
5.4 lies between 5 and 6 hours.
00:16:48.559 --> 00:16:53.279
When the unknown is the exponent, isolate the exponential part first.
00:16:53.679 --> 00:16:59.519
Then take logarithms, use the power rule to bring the exponent down and divide.
00:16:59.679 --> 00:17:02.879
The calculation identifies the exponent precisely.
00:17:03.120 --> 00:17:08.000
1.25 raised to what power produces the required growth factor.
00:17:08.720 --> 00:17:11.440
Now let's try challenge problem number 10.
00:17:11.519 --> 00:17:17.440
And this one's kind of neat because I think it it incorporates all the lessons that we've learned so far today.
00:17:17.680 --> 00:17:28.480
The question is solve the base two logarithm of x minus 1 plus the base 2 logarithm of x minus 3 equals 3.
00:17:28.960 --> 00:17:33.759
And the clue is to be sure to check the domain of both logarithms.
00:17:34.799 --> 00:17:36.000
So what do I know?
00:17:36.240 --> 00:17:38.720
Both logarithms use base 2.
00:17:38.880 --> 00:17:47.920
They are being added, so the product rule may combine them, but before I do any algebra, I need to ask which x values are even allowed.
00:17:48.079 --> 00:17:49.920
So let's set the boundaries.
00:17:50.160 --> 00:17:57.839
A real logarithm can only take a positive input, not zero, not a negative number, positive only.
00:17:58.079 --> 00:17:58.480
Why?
00:17:58.640 --> 00:18:03.759
Because a positive base such as two stays positive when we raise it to any real power.
00:18:04.079 --> 00:18:14.319
Two raised to some real exponent can produce a large positive number or a small positive fraction, but it cannot produce zero or a negative number.
00:18:14.559 --> 00:18:25.039
So for x minus one, that has to be greater than zero, so x has to be one or greater, and x minus three has to be greater than zero, so that gives x greater than three.
00:18:25.279 --> 00:18:26.960
X must be greater than three.
00:18:27.200 --> 00:18:30.240
So both conditions must be true at the same time.
00:18:30.400 --> 00:18:33.839
The stricter condition is x greater than three.
00:18:34.000 --> 00:18:35.119
That's our domain.
00:18:35.279 --> 00:18:39.599
Any answer at or below three is not allowed in the original equation.
00:18:39.759 --> 00:18:43.519
So our domain is x must be greater than three.
00:18:44.000 --> 00:18:47.119
Writing the domain now gives us a boundary to check.
00:18:47.279 --> 00:18:50.079
The algebra may produce more than one candidate later.
00:18:50.240 --> 00:18:54.480
We'll compare each candidate with the original domain to see which ones are valid.
00:18:54.720 --> 00:18:57.359
So now let's combine those two logarithms.
00:18:57.440 --> 00:19:01.119
The logarithms have the same base and they are added.
00:19:01.279 --> 00:19:06.799
The product rule says that a sum of logarithms becomes the logarithm of a product.
00:19:07.039 --> 00:19:09.440
A small number example shows why.
00:19:09.599 --> 00:19:15.759
The base two logarithm of four is two and the base two logarithm of two is one.
00:19:16.240 --> 00:19:18.160
Add them and you get three.
00:19:18.400 --> 00:19:22.240
Multiply the inputs four times two and you get eight.
00:19:22.480 --> 00:19:25.759
The base two logarithm of eight is also three.
00:19:26.000 --> 00:19:32.720
So the base two logarithm of four plus the log base two of two equals two plus one, which is three.
00:19:32.960 --> 00:19:35.599
So the log base two of eight.
00:19:35.839 --> 00:19:38.720
Now apply that rule to the expression in the problem.
00:19:39.039 --> 00:19:46.720
Log base two of, in parentheses, x minus one times x minus three equals three.
00:19:46.960 --> 00:19:49.519
We've not multiplied the expressions out yet.
00:19:49.599 --> 00:19:54.799
We've simply replaced two logarithms with one logarithm of their product.
00:19:55.039 --> 00:19:59.279
Now we translate the raw logarithm into exponential form.
00:19:59.599 --> 00:20:02.559
So let's return to that first master question.
00:20:02.720 --> 00:20:06.880
The base two logarithm of this product equals three.
00:20:07.279 --> 00:20:12.480
That means two raised to the third power equals the product.
00:20:13.039 --> 00:20:20.000
So x minus one times x minus three equals two raised to the third power, which is eight.
00:20:20.240 --> 00:20:20.880
There it is.
00:20:20.960 --> 00:20:22.240
The logarithms are gone.
00:20:22.400 --> 00:20:24.960
What remains is an algebra equation.
00:20:25.200 --> 00:20:27.359
So then we solve the algebra.
00:20:27.599 --> 00:20:30.400
Multiply two expressions on the left.
00:20:30.720 --> 00:20:33.599
x times x is x squared.
00:20:33.759 --> 00:20:41.599
The two middle terms are negative three x and negative one x, which combines to negative four x.
00:20:41.920 --> 00:20:45.200
And negative one times negative three is positive three.
00:20:45.359 --> 00:20:49.519
So x squared minus four x plus three equals eight.
00:20:50.319 --> 00:20:54.000
Subtract eight from both sides so that one side equals zero.
00:20:54.319 --> 00:20:57.920
x squared minus four x minus five equals zero.
00:20:58.160 --> 00:20:59.440
And now we factor.
00:20:59.519 --> 00:21:01.599
This is a quadratic equation.
00:21:01.759 --> 00:21:06.240
We need two numbers that multiply to negative five and add to negative four.
00:21:06.400 --> 00:21:09.519
Those numbers are negative five and positive one.
00:21:10.400 --> 00:21:14.400
A product equals zero when at least one factor equals zero.
00:21:14.480 --> 00:21:22.799
So x minus five equals zero giving x equals five, or x plus one equals zero, giving x equals negative one.
00:21:23.200 --> 00:21:27.680
So our candidates are x equals five or x equals negative one.
00:21:28.160 --> 00:21:29.759
Now let's return to the domain.
00:21:29.920 --> 00:21:32.319
This is where the word candidates matter.
00:21:32.480 --> 00:21:39.599
The quadratic equation produced two values, but our original equation contained logarithms.
00:21:40.000 --> 00:21:42.960
We need to check both values in the original problem.
00:21:43.119 --> 00:21:49.200
Start with x equals five, then x minus one is four, and x minus three is two.
00:21:49.599 --> 00:21:53.839
Both inputs are positive, so both logarithms exist.
00:21:54.079 --> 00:22:00.960
The base two logarithms of four is two, the base two logarithm of two is one, so two plus one equals three.
00:22:01.119 --> 00:22:04.000
So x equals five works.
00:22:04.400 --> 00:22:06.880
Now x equals negative one.
00:22:07.200 --> 00:22:08.400
Let's try that.
00:22:09.039 --> 00:22:13.119
Then x minus one is negative two, and x minus three is negative four.
00:22:13.200 --> 00:22:16.480
In the real number system, those logarithms are not defined.
00:22:16.720 --> 00:22:24.880
Negative one may solve the quadratic we created after combining and translating, but it does not solve the original logarithmic equation.
00:22:25.279 --> 00:22:30.880
x equals negative one, the logarithm inputs are negative, so that's not allowed.
00:22:31.119 --> 00:22:33.839
So the only solution is x equals five.
00:22:33.920 --> 00:22:36.000
So that's our final answer.
00:22:37.119 --> 00:22:41.119
Algebra can produce a number that's not permitted by the original problem.
00:22:41.200 --> 00:22:43.920
With logarithms, check the inputs.
00:22:44.160 --> 00:22:47.359
Every logarithm must receive a positive number.
00:22:47.599 --> 00:22:52.480
Solve the algebra, then check every result in the original equation.
00:22:59.200 --> 00:23:01.759
So that is the logarithms toolkit.
00:23:01.920 --> 00:23:06.880
Start with one question: what exponent produced this number?
00:23:07.039 --> 00:23:27.279
Then use the four-part habit, ask what the problem wants, translate between logarithmic and exponential form, compare values on a logarithmic scale by finding the difference and turning that difference into a factor, solve and then check that answer is allowed in the original problem.
00:23:27.519 --> 00:23:29.599
And remember the two challenge moves.
00:23:29.839 --> 00:23:38.400
If time is hidden in an exponent, isolate the exponential expression and use a logarithm to bring the exponent down.
00:23:38.799 --> 00:23:45.200
If logarithms are added or subtracted, use the product or quotient rule when the bases match.
00:23:45.440 --> 00:23:50.400
And whenever a variable sits inside a logarithm, track the domain.
00:23:50.720 --> 00:23:54.160
You do not need to hold every rule in your head at once.
00:23:54.319 --> 00:23:56.079
Keep the structure in view.
00:23:56.400 --> 00:23:58.240
Base, exponent, result.
00:23:58.960 --> 00:24:03.440
Multiplication underneath becomes addition on the logarithmic scale.
00:24:03.759 --> 00:24:06.319
Division underneath becomes subtraction.
00:24:06.720 --> 00:24:12.720
The equal steps on the printed scale can represent very large ratios in the real world.
00:24:13.119 --> 00:24:18.799
All ten problems are there in the study hall section at studyproblem.org.
00:24:19.119 --> 00:24:26.319
Work through the earlier problems first and then return to problems 9 and 10 and see whether you can explain every step out loud.
00:24:27.200 --> 00:24:34.880
If you can explain why the step is okay and what it means, you understand way more than just the answer.
