Unit 2 · Topic 2.10 · about 30 minutes · 7 parts · 14 scored checks, 2 of them multiple choice

Inverses of Exponential Functions

Built from College Board's course framework · Every answer key independently checked

The graphs of y = 2 to the x and its inverse y = log base 2 of x on equal scales, with the dashed line y = x between them. Points come in swapped pairs: (0, 1) on the exponential and (1, 0) on the logarithm, (1, 2) and (2, 1), (2, 4) and (4, 2), (-1, 1/2) and (1/2, -1). Dotted segments join two of the pairs and cross y = x at right angles, so each curve is the mirror image of the other across that line.

Part 1 Think first

A thousand times bigger

Guess before you learn. A wrong guess still makes the answer stick.

Predict first

A biology class grows a bacteria culture that doubles in size every hour. About how long will it take for the culture to be 1,000 times its starting size?

In short. The logarithmic function $f (x) = \log_{b} x$ is the inverse of the exponential function $g (x) = b^{x}$ , so it takes an output of $b^{x}$ and hands back the input that produced it.

If $(s, t)$ is an ordered pair of $b^{x}$ , then $(t, s)$ is an ordered pair of $\log_{b} x$ , so the graph of $\log_{b} x$ is the graph of $b^{x}$ reflected over the line $y = x$ .
Exponential outputs change multiplicatively as the inputs change additively. Logarithmic outputs change additively as the inputs change multiplicatively.
Each function undoes the other: $b^{\log_{b} x} = x$ and $\log_{b} (b^{x}) = x$ . The general form of a logarithmic function is $f (x) = a \log_{b} x$ , with $b > 0$ , $b \neq 1$ and $a \neq 0$ .

Part 2 Learn it

Four short ideas, each checked as you go

2a. Swap the columns

Start with $g (x) = 2^{x}$ , an exponential function with initial value 1. Here is part of its table.

The exponential function $g (x) = 2^{x}$
$x$	$g (x)$
$- 2$	$\frac{1}{4}$
$- 1$	$\frac{1}{2}$
0	1
1	2
2	4
3	8
4	16

The inverse of $g$ runs this table backward. Give it 8 and it hands back 3, because $g (3) = 8$ . So to build its table, swap the columns. If $(s, t)$ is an ordered pair of $g$ , then $(t, s)$ is an ordered pair of the inverse.

The inverse of $g$ , which is $f (x) = \log_{2} x$
$x$	$f (x)$
$\frac{1}{4}$	$- 2$
$\frac{1}{2}$	$- 1$
1	0
2	1
4	2
8	3
16	4

That inverse is the logarithmic function with base 2, $f (x) = \log_{2} x$ . Each output is the exponent you put on 2 to get the input. You already know $\log_{2} 8 = 3$ ; now you can see it as the row $(8, 3)$ , which is the row $(3, 8)$ of $2^{x}$ turned around.

Quick check · Fill the statement

No calculator. The graph of $g (x) = 3^{x}$ passes through $(4, 81)$ and $(- 2, \frac{1}{9})$ . Tap a blank, then tap an answer.

The graph of $f (x) = \log_{3} x$ passes through the point with input 81 and output , and the point with input and output $- 2$ .

2b. Multiply on one side, add on the other

Read the two tables again, top to bottom.

In the table of $g (x) = 2^{x}$ , the inputs go up by 1 on every row and the outputs are multiplied by 2. Equal-length input intervals give proportional outputs. That is exponential growth: the outputs change multiplicatively as the inputs change additively.

Swapping the columns swaps the pattern. In the table of $f (x) = \log_{2} x$ , the inputs are multiplied by 2 on every row and the outputs go up by 1. Now the inputs change proportionately while the outputs increase over equal-length intervals. That is logarithmic growth: the outputs change additively as the inputs change multiplicatively.

The base sets the size of the multiplying step. For the common logarithm $\log x$ , base 10, the inputs 1, 10, 100 and 1000 give the outputs 0, 1, 2 and 3. Multiplying any input by 10 adds 1 to the output, wherever you are in the table. Multiplying by 100 is two of those steps, so it adds 2.

That is why the culture in Think first needs only about 10 hours to grow a thousandfold. Hours are the output of $\log_{2}$ , and every doubling of the target adds just one hour.

Quick check · Multiple choice

No calculator. For $f (x) = \log_{4} x$ , an input is multiplied by 16. What happens to the output of $f$ ?

2c. Each one undoes the other

When $b > 0$ and $b \neq 1$ , the functions $f (x) = \log_{b} x$ and $g (x) = b^{x}$ are inverse functions. Whatever one does, the other undoes, in either order:

g (f (x)) = f (g (x)) = x

Written out, that is $b^{\log_{b} x} = x$ and $\log_{b} (b^{x}) = x$ .

Check it on the base 2 tables. $f (8) = 3$ , and then $g (3) = 8$ , so you are back at 8. The other order: $g (3) = 8$ , and then $f (8) = 3$ , back at 3.

This lets you simplify without knowing what the logarithm equals. $\log_{5} 7$ is not a whole number, but $5^{\log_{5} 7} = 7$ , because 5 raised to the exponent that gives 7 is 7. In the same way, $\log (10^{2.6}) = 2.6$ .

The bases have to match. In $2^{\log_{3} 9}$ the logarithm is base 3 and the power is base 2, so nothing cancels: $\log_{3} 9 = 2$ , and $2^{2} = 4$ , not 9.

Quick check · Match

No calculator. Tap an expression, then its value.

2d. The mirror line

Plot the points of $g (x) = 2^{x}$ and of its inverse $f (x) = \log_{2} x$ on the same axes, as in the diagram. Each point $(s, t)$ on one graph has a partner $(t, s)$ on the other: $(2, 4)$ and $(4, 2)$ , $(- 1, \frac{1}{2})$ and $(\frac{1}{2}, - 1)$ .

Swapping the coordinates of a point reflects it over the line $y = x$ , the graph of the identity function $h (x) = x$ . So the graph of $f (x) = \log_{b} x$ is the graph of $g (x) = b^{x}$ reflected over $y = x$ . Fold the picture along that line and the two curves land on each other. The point $(0, 1)$ of every $b^{x}$ lands on $(1, 0)$ , a point of every $\log_{b} x$ .

Not every logarithmic function is a plain $\log_{b} x$ . The general form is $f (x) = a \log_{b} x$ , with base $b$ , where $b > 0$ , $b \neq 1$ and $a \neq 0$ . The constant $a$ multiplies every output. For $3 \log_{2} x$ , the inputs 1, 2, 4 and 8 give 0, 3, 6 and 9: the inputs still multiply by the same factor, and the outputs still rise in equal steps, now of 3.

Quick check · Spot the error

No calculator. A student sketches $f (x) = \log_{3} x$ from the graph of $g (x) = 3^{x}$ . Tap the line with the mistake.

Part 3 Watch it done

Building the inverse from a table

Watch each step, then do the last one yourself.

Let $g (x) = 4^{x}$ , and let $f$ be the inverse of $g$ . Build a table for $f$ , write $f (x)$ , and find $f (8)$ .

Step 1 · Table the exponential
Use inputs that go up by 1: $g (- 2) = \frac{1}{16}$ , $g (- 1) = \frac{1}{4}$ , $g (0) = 1$ , $g (1) = 4$ , $g (2) = 16$ and $g (3) = 64$ .
Step 2 · Swap the columns
Each pair $(s, t)$ of $g$ becomes $(t, s)$ for $f$ . So $f$ takes the inputs $\frac{1}{16}$ , $\frac{1}{4}$ , 1, 4, 16 and 64 to the outputs $- 2$ , $- 1$ , 0, 1, 2 and 3. That is $f (x) = \log_{4} x$ .
Why this step?
Why is swapping the columns enough to make $f$ the inverse of $g$ ?
Because then $f$ undoes $g$ on every row. $g$ sends 3 to 64 and $f$ sends 64 back to 3, so $f (g (3)) = 3$ , and the same is true of each pair in the table.
Step 3 · Place 8 in the table
Down the table of $f$ , each input is 4 times the one before and each output is 1 more. The input 8 is not listed. It sits between 4 and 16, so $f (8)$ sits between $f (4) = 1$ and $f (16) = 2$ .
Your step · Calculate

No calculator. Finish it. What is $f (8)$ ? Give a decimal.

Your answer

Part 4 Try it

How many digits for the lock?

A keypad lock uses a code of $n$ digits, each from 0 to 9. It has $N (n) = 10^{n}$ possible codes, so a 4-digit lock has 10,000 of them. A lock maker works the other way round. She starts from how many possible codes she wants and needs the number of digits that gives exactly that many. That is a question for the inverse of $N$ .

Try it · Decide and justify

No calculator. Which function $D$ takes a number of possible codes $x$ and returns the number of digits that gives exactly that many codes?

Which reason supports your answer?

Try it · Calculate

No calculator. The lock maker wants exactly 1,000,000 possible codes. Use the inverse of $N (n) = 10^{n}$ . How many digits does the code need?

Your answer

Part 5 Explain it

Write it the way the exam asks

Nothing you type leaves this page.

Free response is 37.5% of the exam score. This one is shaped like Question 1, Function Concepts.

Free-response practice · Question 1 style

Graphing calculator required. The function $g$ is given by $g (x) = 5^{x}$ , and $f$ is the inverse of $g$ . (i) Write an expression for $f (x)$ . (ii) The point $(3, 125)$ is on the graph of $g$ . Identify the corresponding point on the graph of $f$ , and describe how the graph of $f$ is related to the graph of $g$ . (iii) The inputs 1, 5, 25 and 125 are each 5 times the one before. Describe how the corresponding outputs of $f$ change. (iv) Find the value of $g (f (40))$ , and explain your reasoning.

Your answer stays on this page. It is never sent or saved.

Model answer

(i) $f (x) = \log_{5} x$ . (ii) The point $(125, 3)$ is on the graph of $f$ , because an inverse swaps the input and output of every ordered pair. The graph of $f$ is the reflection of the graph of $g$ over the line $y = x$ . (iii) The outputs are 0, 1, 2 and 3, so they go up by 1 each time the input is multiplied by 5. (iv) $g (f (40)) = 40$ . Since $f$ and $g$ are inverse functions, $g$ undoes $f$ : $5^{\log_{5} 40} = 40$ .

Score your own answer. This is a self-check, not a grade. Tick each point your answer makes.

Writes $f (x) = \log_{5} x$ .
Identifies $(125, 3)$ and says the graph of $f$ is the graph of $g$ reflected over $y = x$ .
Describes the outputs of $f$ as increasing by 1 each time the input is multiplied by 5.
Gives $g (f (40)) = 40$ , explaining that $g$ undoes $f$ because they are inverse functions.

Part 6 Check yourself

One table, two functions

The two case questions use the same table. Answer them from it.

The table gives selected values of an exponential function $g (x) = b^{x}$ . The function $f$ is the inverse of $g$ .

Selected values of $g$
$x$	$g (x)$
$- 1$	0.4
0	1
1	2.5
2	6.25
3	15.625

Case question 1 · Multiple choice

No calculator. Which point lies on the graph of $f$ ?

Case question 2 · Calculate

No calculator. Each output of $g$ in the table is 2.5 times the one before. What is $f (97.65625)$ ?

Your answer

The rest stand on their own and do not use the case.

Check yourself · Sort it

No calculator. Each card lists four inputs and their outputs. Tap a card, then the pattern it shows.

Exponential pattern

Logarithmic pattern

Neither

Check yourself · Spot the error

No calculator. A student simplifies four expressions, one per line. Tap the line with the mistake.

Check yourself · Select all

No calculator. Which two of these are logarithmic functions in general form? Pick two.

Select 2 answers, then check.

Part 7 Keep it

Bring back earlier ideas

Pulling an older idea back up is what makes it last.

From 2.5 Exponential Function Context and Data Modeling · Fill the statement

No calculator. A town's population grows by 3% each year. One year after the count began, it had 12,360 people. Tap a blank, then tap an answer.

The model is $P (t) = a \cdot b^{t}$ , with $t$ in years, where $a$ = and $b$ = .

From 2.2 Change in Linear and Exponential Functions · Decide and justify

No calculator. A function $f$ has $f (1) = 12$ , $f (2) = 18$ , $f (3) = 27$ and $f (4) = 40.5$ . Based on these values, what kind of function is $f$ ?

Which reason supports your answer?

In three lines

The inverse of $b^{x}$ is $\log_{b} x$ : every ordered pair $(s, t)$ of $b^{x}$ becomes $(t, s)$ , and the graph reflects over $y = x$ .
Exponential outputs multiply as the inputs add; logarithmic outputs add as the inputs multiply.
$b^{\log_{b} x} = x$ and $\log_{b} (b^{x}) = x$ when the bases match, and the general form is $a \log_{b} x$ with $b > 0$ , $b \neq 1$ , $a \neq 0$ .

Vocabulary

Logarithmic function: A function of the form $f (x) = a \log_{b} x$ , with base $b$ , where $b > 0$ , $b \neq 1$ and $a \neq 0$ .
Inverse functions: Two functions that undo each other, as $f (x) = \log_{b} x$ and $g (x) = b^{x}$ do: $g (f (x)) = f (g (x)) = x$ .
Identity function: The function $h (x) = x$ , whose graph is the line over which the graphs of $b^{x}$ and $\log_{b} x$ reflect onto each other.
Exponential growth: Outputs changing multiplicatively as the inputs change additively, as in a table of $b^{x}$ .
Logarithmic growth: Outputs changing additively as the inputs change multiplicatively, as in a table of $\log_{b} x$ .

Course alignment, for teachers

AP Precalculus topic 2.10, Unit 2: Exponential and Logarithmic Functions.

Skill 1.C: Construct new functions, using transformations, compositions, inverses, or regressions, that may be useful in modeling contexts, criteria, or data, with and without technology.
Skill 2.B: Construct equivalent graphical, numerical, analytical, and verbal representations of functions that are useful in a given mathematical or applied context, with and without technology.

This lesson maps all 5 essential knowledge statements of the topic to the part that teaches each one: 5 are checked by a scored question. The map is in the lesson file.