2.11 Logarithmic Functions

Logarithms as inverses of exponentials

A logarithmic function is the inverse of an exponential function: if y = b^x, then x = log_b(y). Since exponential functions are one-to-one, their inverses exist and are also functions. The domain of log_b(x) is x > 0, and its range is all real numbers — the opposite of the exponential's domain/range.

Key features of log graphs

For f(x) = log_b(x) with b > 1, the graph increases, passes through (1, 0), has a vertical asymptote at x = 0, and rises slowly (concave down). As x → 0⁺, f(x) → -∞. If 0 < b < 1, the function decreases instead. Logarithmic growth is much slower than linear or exponential growth as x increases.

Analyzing rate of change

Because logarithmic functions grow ever more slowly, their average rate of change decreases as x increases, though it stays positive (for b > 1) — the function is always increasing but at a decreasing rate. This concave-down, increasing behavior contrasts with the concave-up, increasing behavior of exponential functions.

Key terms

Logarithmic function definition

The inverse of an exponential function: y = b^x is equivalent to x = log_b(y).

Domain of log_b(x)

x > 0

Range of log_b(x)

All real numbers

Vertical asymptote of log_b(x)

x = 0

Shape of y = log_b(x), b > 1

Increasing, concave down, passing through (1, 0), with rate of change decreasing but staying positive.

Try an AP-style question

Which describes y = log₃ x?

1. Increasing and concave up
2. Decreasing and concave up
3. Increasing and concave down
4. Decreasing and concave down