Lecture: Logarithmic Functions
Lecture script · about 15–20 minutes spoken · Week 14
Opening
A sound engineer reads a decibel meter. A lab technician counts how many times a sample was cut in half. Both are asking how many times something was multiplied. That question has a tool built for it, the logarithm, and it is the subject of this lecture.
Here's where we are in this course. In Week 1 you learned the exponent rules and the powers of two, from 1 up to 1024. In Week 3 you met domain, the set of inputs a rule accepts. In Week 10 you met the vertical asymptote. And last week, in Week 13, you built exponential functions such as 3 times 2 to the x. This week we run that exponential backwards. Next week, in Week 15, you'll use what we build here to solve equations for an unknown exponent.
I'll take eight ideas in order, from what a logarithm is to the one move that has no rule.
1. Ask the exponent question
Subtracting undoes adding, and halving undoes doubling. So what undoes raising a base to a variable exponent?
Start with last week's function, 3 times 2 to the x. Put in 3 and you get 3 times 8, which is 24. Now flip it around. Suppose I tell you the output is 24 and ask for the input. First, peel off the 3. It multiplies the power, so divide both sides by 3, and you're left with 2 to the x equals 8. Now the real question: which power of 2 is 8? Two times 2 times 2 is 8, so the power is 3. Check it: 3 times 2 cubed is 3 times 8, which is 24. It works.
That middle question, "which power of 2 is 8," comes up so often that it gets its own name. A logarithm is the exponent you need: log base b of x equals y means b to the y equals x. So log base 2 of 8 is 3, because 2 cubed is 8. The base is positive and not 1, and the number inside must be positive.
A lab culture starts with 5 cells and doubles every hour. When does it reach 40 cells? Divide 40 by 5 and you get 8. Which power of 2 is 8? It's 3. So the culture reaches 40 cells after 3 hours.
If you remember one sentence from this lecture, make it this one: the log is the exponent you need.
Pause and try this. In 3 times 2 to the x equals 24, is it fair to combine 3 times 2 into 6 first and ask which power of 6 is 24?
Give it a moment before you read on.
Here's the answer. No, it isn't fair, because the exponent belongs to the 2 alone; 3 times 2 cubed is 24, while 6 cubed is 216.
2. Turn log form into exponential form and back
Think of a recipe card and a shopping list. They describe the same meal, laid out two ways. A logarithm and a power are two layouts of the same fact.
Exponential form is a statement written with a power, such as 4 cubed equals 64. Logarithmic form is a statement written with a logarithm, such as log base 4 of 64 equals 3. It names an exponent as the output.
Two things never change as you switch between them. The base stays the base. And the value of the log is always the exponent. So log base 4 of 64 equals 3 turns into 4 to the 3rd equals 64. Check the arithmetic: 4 times 4 is 16, and 16 times 4 is 64.
Going the other way works the same. Ten squared is 100, so log base 10 of 100 is 2. Five to the zero is 1, so log base 5 of 1 is 0.
Pause and try this. Someone rewrites log base 4 of 64 equals 3 as 3 to the 4th equals 64. Does that rewrite pass its own check?
Take a breath and test it before you go on.
Here's the answer. It fails, because 3 to the 4th is 81, not 64; the base 4 has to stay the base.
3. Read a logarithm in your head
Most logs this week need no calculator, only the powers of two.
Read them backwards, and they become a table of logs. Two to the zero is 1, so log base 2 of 1 is 0. Two cubed is 8, so log base 2 of 8 is 3. Two to the 6th is 64, so log base 2 of 64 is 6. And 2 to the 10th is 1024, so log base 2 of 1024 is 10. A video shared by 1 person that doubles its shares each hour reaches 1024 shares after 10 hours, and 10 is exactly log base 2 of 1024.
What about numbers smaller than 1? Week 1's negative exponents answer that. Two to the negative 2 is 1 over 2 squared, which is 1 over 4. So log base 2 of one fourth is negative 2. A negative log means a reciprocal of a power, never a negative number inside.
Two values come out the same in every base. Any base to the zero is 1, so the log of 1 is always 0. Any base to the first is itself, so the log of the base is always 1.
A 100-gram sample halves every period: 100, then 50, then 25, then 12.5 grams. Someone hands you the jar at 12.5 grams. How many half-lives have passed? Divide 12.5 by 100 and you get one eighth. A half, raised to some power, gives one eighth, which means 2 to that power gives 8. That power is log base 2 of 8, which is 3. Three half-lives.
Pause and try this. What is log base 5 of 1?
Think about which power of 5 gives you 1.
Here's the answer. It's 0, because 5 to the zero is 1, and the log of 1 is 0 in every base.
4. Use the common and natural logarithms
Two bases show up so often that they get short names. One comes from the way we count in tens. The other is the natural base, e.
The common logarithm is the logarithm with base 10, written log of x with no base shown. So log of 1000 is 3, because 10 cubed is 1000. The natural logarithm is the logarithm with base e, the constant near 2.71828, written ln of x. So ln of e is 1, because e to the first is e. And ln of 1 is 0, because e to the zero is 1.
Some measuring scales are built on a logarithm of a ratio. The decibel scale for sound and the pH scale for acidity work this way. On a plain base-10 scale, a reading 1000 times the reference sits 3 steps up, because log of 1000 is 3. A reading 100 times the reference sits 2 steps up. The two readings differ by one step, but 1000 versus 100 is a tenfold change. On a base-10 scale, one step up means ten times as much, not one more unit.
Pause and try this. A friend says log base 10 of 100 is 10, since 100 divided by 10 is 10. Is that right?
Ask yourself which question a log is really asking.
Here's the answer. It's not right, because a log asks for an exponent, and 10 squared is 100, so log base 10 of 100 is 2.
5. Draw the logarithm as the inverse of the exponential
Picture a coat check. You hand over a coat and get a ticket. Later, the ticket brings the coat back. The second counter runs the first one in reverse.
An inverse function is a function that undoes another one by swapping its inputs and outputs. The function log base 2 of x undoes 2 to the x. A logarithmic function is a function of the form log base b of x. Its domain is the positive numbers, and its range is all real numbers.
Let's build the graph from points. The curve 2 to the x passes through the point negative 2 comma one fourth, the point 0 comma 1, the point 1 comma 2, and the point 3 comma 8. Swap each pair. You get one fourth comma negative 2, then 1 comma 0, then 2 comma 1, then 8 comma 3. Those are points on the log curve. The point 8 comma 3 says log base 2 of 8 is 3, which matches our table.
Now describe the whole log curve in words. It lives only to the right of the vertical axis, because you can only take the log of a positive number. It crosses the horizontal axis at 1, since log base 2 of 1 is 0. As the input shrinks toward 0, the curve drops lower and lower, hugging the vertical axis without ever touching it. That axis, the line x equals 0, is its vertical asymptote. And it rises slowly: from an input of 1 to an input of 1024, the output climbs only from 0 to 10.
The diagonal line y equals x is the mirror between the two curves.
Pause and try this. The point 3 comma 8 sits on the curve 2 to the x. Which point does it become on the curve log base 2 of x?
Swap the two numbers in your head.
Here's the answer. It becomes 8 comma 3, because the inverse trades inputs and outputs, and log base 2 of 8 is 3.
6. Find a domain and a vertical asymptote
A log accepts only positive numbers inside it. The thing inside the parentheses is called the argument of a logarithm. When the argument has x in it, the domain rule turns into a Week 2 inequality.
Take log base 3 of x minus 2. The argument, x minus 2, must be greater than 0. Add 2 to both sides, and x must be greater than 2. So the domain is the interval from 2 to infinity, with 2 left out. The argument is zero at x equals 2, so the vertical asymptote is the line x equals 2. Check a couple of points. At x equals 3, you get log base 3 of 1, which is 0. At x equals 11, you get log base 3 of 9, which is 2, because 3 squared is 9.
For log of x plus 5, you need x plus 5 greater than 0, so x is greater than negative 5. The domain starts at negative 5, and the asymptote is x equals negative 5. For ln of 2x minus 1, you need 2x minus 1 greater than 0. Add 1, then divide by 2, and x must be greater than one half. The asymptote is x equals one half.
And what about log base 2 of negative 8? It would need 2 to some power to equal negative 8. Every power of 2 is positive, so there's no real value at all.
Pause and try this. A classmate says log base 2 of negative 8 is negative 3. How would you test that claim?
Try writing it as a power of 2 first.
Here's the answer. Rewrite it as 2 to the negative 3, which is one eighth, a positive number, so the claim fails and log base 2 of negative 8 has no real value.
7. Use the three properties
Here's a pattern you've already seen. Eight times 4 is 32. In powers of two, that's 2 cubed times 2 squared equals 2 to the 5th, because the exponents add: 3 plus 2 is 5. A log is an exponent, so multiplying inside a log should become adding outside it.
That's the product property of logarithms: the log of a product is the sum of the logs. Check it: log base 2 of 8 times 4 is log base 2 of 32, which is 5. And log base 2 of 8 plus log base 2 of 4 is 3 plus 2, which is 5.
The quotient property of logarithms says the log of a quotient is the difference of the logs. Log base 2 of 64 over 4 is log base 2 of 16, which is 4. And 6 minus 2 is 4.
The power property of logarithms says the exponent inside the log comes down in front as a multiplier. Log base 2 of 8 squared is log base 2 of 64, which is 6. And 2 times log base 2 of 8 is 2 times 3, which is 6.
Each property is a Week 1 exponent rule wearing a logarithm.
Pause and try this. Is log base 2 of 8 squared the same as log base 2 of 8, times itself?
Run both versions with the numbers from our table.
Here's the answer. They differ, because log base 2 of 8 times itself is 3 times 3, or 9, while log base 2 of 64 is 6.
8. Expand and condense: no rule for a sum
Packing a suitcase and unpacking it use the same items in opposite directions. Expanding and condensing are those two moves for logarithms.
To expand a logarithm is to rewrite one log of a product, quotient, or power as a sum or difference of simpler logs. Take log base b of x cubed times y squared. The product property splits it into two logs, and the power property brings each exponent down. You get 3 times log base b of x, plus 2 times log base b of y. Let's check it with numbers. Use base 2, with x equal to 2 and y equal to 4. Then x cubed times y squared is 8 times 16, which is 128, and log base 2 of 128 is 7. The expanded side gives 3 times 1, plus 2 times 2, which is 3 plus 4, or 7. They match.
To condense logarithms is to combine a sum or difference of logs with the same base into a single log. Take 2 log of x, plus log of y. Push the 2 up into an exponent first, then merge. You get the log of x squared times y. Check with x and y both equal to 10. The left side is 2 plus 1, which is 3. The right side is log of 1000, which is also 3.
Now the trap the whole week has been building toward. There's a rule for a product, a quotient, and a power. There is no rule for the log of a sum. Watch. Log base 2 of 4 plus 4 is log base 2 of 8, which is 3. But log base 2 of 4, plus log base 2 of 4, is 2 plus 2, which is 4. Three is not 4, so the split fails. A sum of logs does mean something: it's the log of a product. So log of x plus log of 4 is log of 4x, never log of x plus 4.
Pause and try this. Does log base 2 of 4 plus 4 equal log base 2 of 4 plus log base 2 of 4?
Add first on one side, and split on the other.
Here's the answer. It doesn't, because adding first gives log base 2 of 8, which is 3, while splitting gives 2 plus 2, which is 4.
Closing
Let's pull it together in the order we took it. A logarithm is the exponent you need, so log base 2 of 8 is 3. Logarithmic form and exponential form say the same thing, and the base stays the base. You can read many logs from the powers of two, the log of 1 is always 0, and a reciprocal gives a negative log. The common log has base 10, and the natural log has base e. The log curve is the exponential curve with every point swapped, with a vertical asymptote where the argument is zero. Set the argument greater than zero to find the domain. Product becomes sum, quotient becomes difference, and a power becomes a multiplier. And there is no rule for the log of a sum.
Next in this module, you'll find the slide deck, then the Readings & Resources page, the lecture tutorial, the practice set, and the quiz. Next week, in Week 15, Exponential & Logarithmic Equations & Applications, these properties become your method for solving for an exponent.
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