AP® Precalculus review sheet from Aim for Five (aimforfive.com/precalc/units/2/2-10)
Unit 2 · Topic 2.10
2.10 Inverses of Exponential Functions
The logarithmic function log_b(x) is the inverse of the exponential function bˣ. Their input-output pairs swap, their graphs are reflections over y = x, and the exponential's horizontal asymptote becomes the log's vertical asymptote.
Key terms
- inverse
- logarithmic function
- vertical asymptote
- reflection
The general log function
A logarithmic function in general form is f(x) = a · log_b(x), with b > 0, b ≠ 1 and a ≠ 0.
g(x) = bˣ and f(x) = log_b(x) are inverse functions. That means g(f(x)) = b^(log_b x) = x for x > 0, and f(g(x)) = log_b(bˣ) = x for every real x.
The constant a stretches the graph vertically. With a > 0 and b > 1, a · log_b(x) is increasing. A negative a flips the graph over the x-axis, which topic 2.11 explores.
Swapped pairs, reflected graphs
If (s, t) is on the graph of y = bˣ, then (t, s) is on the graph of y = log_b(x).
For base 2: the exponential has points (−1, 1/2), (0, 1), (1, 2), (3, 8), so the log has points (1/2, −1), (1, 0), (2, 1), (8, 3).
Graphically, y = log_b(x) is the reflection of y = bˣ over the line y = x. The exponential's horizontal asymptote y = 0 reflects into the log's vertical asymptote x = 0. The exponential's domain (all reals) becomes the log's range, and the exponential's range (positive numbers) becomes the log's domain.
Additive vs multiplicative patterns
Exponential functions and log functions change in opposite ways:
- Exponential: when the inputs increase by equal amounts (adding), the outputs change by equal factors (multiplying).
- Logarithmic: when the inputs change by equal factors (multiplying), the outputs change by equal amounts (adding).
| x | 1 | 2 | 4 | 8 | 16 |
|---|---|---|---|---|---|
| log₂(x) | 0 | 1 | 2 | 3 | 4 |
Spotting a log pattern in data
In the table above, each input is double the one before, and each output is 1 more than the one before. Doubling inputs, adding to outputs: that's logarithmic.
So when a table's inputs grow by a constant ratio and its outputs grow by a constant difference, a log model fits. The constant ratio of the inputs tells you the base, after adjusting for any vertical stretch.
Undoing each other in practice
An exponential and a log with the same base cancel each other. log₅(5⁷) = 7, and 3^(log₃ 20) = 20.
You'll use this constantly in topic 2.13. To free an x that's stuck in an exponent, take a log. To free an x that's stuck inside a log, exponentiate both sides with the same base.
It also lets you read log values off an exponential graph. Since (5, 32) is on y = 2ˣ, the point (32, 5) is on y = log₂(x), so log₂(32) = 5.
Worked examples
Try each one yourself first, then open the solution.
- Example 1
Points on an exponential and its inverse
The points (−1, 1/3), (0, 1), (1, 3) and (2, 9) are on y = 3ˣ. Give four points on y = log₃(x), and state that function's domain, range and asymptote.
Show the solutionHide the solution
- Step 1: Swap each pair: (1/3, −1), (1, 0), (3, 1), (9, 2).
- Step 2: 3ˣ has domain all reals and range y > 0, so log₃(x) has domain x > 0 and range all reals.
- Step 3: The horizontal asymptote y = 0 of 3ˣ becomes the vertical asymptote x = 0 of log₃(x).
Answer: (1/3, −1), (1, 0), (3, 1), (9, 2); domain x > 0, range all real numbers, vertical asymptote x = 0.
- Example 2
Building a log model from a table
A function has these values: f(1) = 2, f(4) = 5, f(16) = 8, f(64) = 11. Show that a logarithmic function fits and find it.
Show the solutionHide the solution
- Step 1: Inputs: each is 4 times the previous one (constant ratio).
- Step 2: Outputs: each is 3 more than the previous one (constant difference). Multiplying inputs while adding to outputs is the log pattern.
- Step 3: Every time x is multiplied by 4, the output rises by 3. So f(x) = 2 + 3 log₄(x), where the 2 comes from f(1) = 2 (since log₄(1) = 0).
- Step 4: Check: f(16) = 2 + 3 · 2 = 8 and f(64) = 2 + 3 · 3 = 11.
Answer: f(x) = 2 + 3 log₄(x).
- Example 3
Trap: the inverse identities need the right domain
Let f(x) = 10ˣ and g(x) = log(x). Find f(g(100)), g(f(−2)) and f(g(−5)).
Show the solutionHide the solution
- Step 1: f(g(100)) = 10^(log 100) = 10² = 100.
- Step 2: g(f(−2)) = log(10⁻²) = −2. Negative inputs are fine here, because 10⁻² is a positive number.
- Step 3: f(g(−5)) needs log(−5) first, which is undefined. The identity 10^(log x) = x only holds for x > 0.
Answer: f(g(100)) = 100; g(f(−2)) = −2; f(g(−5)) is undefined.
Common mistakes
- Swapping the asymptotes incorrectly. The exponential has a horizontal asymptote y = 0; the log has a vertical asymptote x = 0.
- Thinking log functions grow by a constant amount for equal input steps. They add a constant for equal input ratios.
- Applying b^(log_b x) = x to zero or negative x.
On the exam
- Expect questions that give points on an exponential function and ask for points on its inverse, or that ask which table shows logarithmic behavior.
- When justifying a log model from data, say: “as the inputs change proportionally, the outputs change by a constant amount.”
Connected topics
Videos
Check yourself
4 questions on 2.10 Inverses of Exponential Functions. Pick an answer to see if you got it, and why.
Let f(x) = 2ˣ, and let g be the inverse function of f. Which of the following is true?
The graph of f(x) = bˣ, where b > 0, passes through the point (3, 64). Which of the following points is on the graph of f⁻¹?
Let g(x) = 3ˣ − 2. The graph of g has a horizontal asymptote at y = −2. Which of the following is true about the graph of g⁻¹?
Let f(x) = 2 + 5ˣ, which has domain all real numbers and range y > 2. Which of the following gives the domain and range of f⁻¹?
0 of 4 answered