2.10 Inverses of Exponential Functions

Logarithms are inverse exponentials

A logarithmic function is the inverse of an exponential function. If y = b^x (b > 0, b ≠ 1), then its inverse is x = b^y, written as y = log_b(x). This means log_b(x) answers the question: 'to what power must b be raised to get x?' Since exponentials are one-to-one, their inverses are true functions.

Domain, range, and graph swap

Because inverse functions reflect over y = x, the domain and range switch: exponential functions have domain all reals and range (0, ∞); logarithmic functions have domain (0, ∞) and range all reals. The exponential's horizontal asymptote (y = 0) becomes the log's vertical asymptote (x = 0), shifted with any transformations.

Key inverse properties

Since log and exponential functions undo each other: b^(log_b x) = x and log_b(b^x) = x. Also, log_b(1) = 0 and log_b(b) = 1 always. These identities let you solve exponential and logarithmic equations by rewriting one form as the other.

Key terms

Logarithmic function

The inverse of an exponential function; y = log_b(x) means b^y = x.

Domain of y = log_b(x)

(0, ∞) — inherited from the range of the exponential function.

Range of y = log_b(x)

All real numbers — inherited from the domain of the exponential function.

Asymptote transformation

The exponential's horizontal asymptote (y = 0) becomes the log's vertical asymptote (x = 0).

Inverse property identities

b^(log_b x) = x and log_b(b^x) = x, since exponential and log functions undo each other.

Try an AP-style question

What is the inverse of f(x) = 2ˣ?

1. log₂ x
2. 2/x
3. log x / 2
4. x²