2.9 Logarithmic Expressions

Logarithms undo exponentials

A logarithm answers: 'what power?' The expression log_b(x) = y means b^y = x, where b is the base (b > 0, b ≠ 1) and x > 0. Logs and exponentials are inverse functions, so their graphs reflect across the line y = x. Common log (base 10) and natural log (ln, base e) are used most often.

Properties of logarithms

Because logs are inverse to exponents, they convert multiplication into addition: log_b(mn) = log_b(m) + log_b(n), log_b(m/n) = log_b(m) − log_b(n), and log_b(m^p) = p·log_b(m). These rules let you expand or condense logarithmic expressions, which is essential for solving equations and rewriting data relationships.

Change of base

Calculators only compute base-10 or base-e logs, so use the change of base formula: log_b(x) = log(x)/log(b) = ln(x)/ln(b). This lets you evaluate a logarithm of any base and compare growth rates between different exponential functions.

Key terms

log_b(x) = y

Means b^y = x; the logarithm gives the exponent needed to produce x from base b.

Product rule

log_b(mn) = log_b(m) + log_b(n)

Quotient rule

log_b(m/n) = log_b(m) − log_b(n)

Power rule

log_b(m^p) = p·log_b(m)

Change of base formula

log_b(x) = ln(x)/ln(b) = log(x)/log(b), used to evaluate logs of any base with a calculator.

Try an AP-style question

log₂ 32 = ?

1. 4
2. 5
3. 16
4. 6