AP PRECALCULUS — UNIT 2B · EXPONENTIAL & LOGARITHMIC FUNCTIONS
2.10 — Inverses of Exponential Functions
Notes — Logarithms as Function Inverses
💡 Learning Objectives (2.10.A)
By the end of this lesson you will be able to:
- Recognize that f(x) = b^x and g(x) = log_b(x) are inverse functions
- Use the composition rules log_b(b^x) = x and b^(log_b(x)) = x to simplify expressions
- Find the inverse of a given exponential function algebraically
- Describe the relationship between the graphs of f and f⁻¹ as a reflection across y = x
1. What ‘Inverse’ Means
Two functions f and g are inverses of each other if applying one undoes the other. Symbolically, f(g(x)) = x for every x in the domain of g, and g(f(x)) = x for every x in the domain of f. The notation for the inverse of f is f⁻¹ — note that the −1 is NOT an exponent here, it is a label.
⚠️ Common mistake
f⁻¹(x) does not mean 1/f(x). The superscript −1 in this context is purely notational, indicating ‘inverse of f’ — not a reciprocal.
2. The Logarithm Is the Inverse of the Exponential
Recall from 2.9 that log_b(y) = x means exactly that b^x = y. This is the same as saying ‘the input that gives an output of y under f(x) = b^x is x = log_b(y).’ That is the very definition of an inverse function.
If f(x) = b^x then f⁻¹(x) = log_b(x).
3. Composition Identities
Because exponentials and logarithms undo each other, composing them in either order returns the input:
- log_b(b^x) = x for every real x
- b^(log_b(x)) = x for every x > 0
📘 Example — Using composition rules
- log₅(5⁷) = 7
- 3^(log₃(11)) = 11
- ln(e^(2x − 1)) = 2x − 1
- e^(ln(7)) = 7
4. Finding an Inverse Algebraically
To find the inverse of an exponential function f(x), follow this procedure:
- Replace f(x) with y.
- Swap x and y in the equation.
- Solve the new equation for y.
- Replace y with f⁻¹(x).
📘 Example — Inverse of f(x) = 2 · 3^x
- Start: y = 2 · 3^x
- Swap: x = 2 · 3^y
- Isolate the exponential: x/2 = 3^y
- Convert to log form: y = log₃(x/2)
- Result: f⁻¹(x) = log₃(x/2)
5. Domain and Range Swap
Because inverses swap inputs and outputs, the domain and range trade places too.
f(x) = b^x | f⁻¹(x) = log_b(x) | |
|---|---|---|
Domain | all real numbers | x > 0 |
Range | y > 0 | all real numbers |
Key point | (0, 1) | (1, 0) |
The horizontal asymptote of the exponential becomes the vertical asymptote of the logarithm, and vice versa.
6. Graphical Reflection
The graph of f⁻¹ is the reflection of the graph of f across the line y = x. Each point (a, b) on f corresponds to a point (b, a) on f⁻¹. So if (2, 9) lies on f(x) = 3^x, then (9, 2) lies on f⁻¹(x) = log₃(x).
7. Summary
- Logarithmic functions are the inverses of exponential functions with the same base
- log_b(b^x) = x and b^(log_b(x)) = x for valid inputs
- Find an inverse by swapping x and y and solving for y
- Domain and range swap; the graph reflects across y = x
- f⁻¹(x) is a label, not a reciprocal