Unit 2: Exponential & Logarithmic Functions

Logarithm Properties & Equations

Using the product, quotient, and power properties of logarithms to rewrite expressions and to solve exponential and logarithmic equations and inequalities.

Advanced25 min lesson4 min readUpdated September 11, 2026Author not yet attributed

Prerequisites

Logarithms as Inverses

Turning Products into Sums, and Solving for a Hidden Exponent

The previous lesson used a single logarithm to undo a single exponential. Real problems rarely stay that tidy: an unknown can sit inside a product, a quotient, or an exponent buried in a larger expression. Before reading on, consider log(8·x) — is there a way to pull the known part, log(8), out of the way so the unknown stands alone?

Definition — Properties of Logarithms

For a base b > 0 (b ≠ 1) and positive inputs M and N: the product property says log_b(MN) = log_b(M) + log_b(N); the quotient property says log_b(M/N) = log_b(M) − log_b(N); and the power property says log_b(M^p) = p·log_b(M). Each mirrors an exponent law — logarithms convert multiplication to addition, division to subtraction, and a power to a coefficient.

\log_b(MN) = \log_b M + \log_b N \qquad \log_b\!\left(\tfrac{M}{N}\right) = \log_b M - \log_b N \qquad \log_b(M^p) = p\,\log_b M

A fourth identity, the change-of-base formula, rewrites any logarithm with a base a calculator actually has: log_b(x) = ln(x) / ln(b) = log(x) / log(b). It follows from taking ln of both sides of b^(log_b x) = x and applying the power property.

\log_b x = \frac{\ln x}{\ln b} = \frac{\log x}{\log b}

Worked Example — Expanding a Single Logarithm

Write log(7x³/y) as a sum and difference of simpler logarithms. The quotient property splits the fraction: log(7x³) − log(y). The product property splits the numerator: log(7) + log(x³) − log(y). The power property lowers the exponent: log(7) + 3·log(x) − log(y). Every factor now appears as its own term, with a coefficient instead of an exponent.

Worked Example — Condensing a Logarithmic Expression

Write 2·ln(x) − ln(x + 1) as a single logarithm. The power property turns each coefficient back into an exponent: ln(x²) − ln(x + 1). The quotient property recombines the difference: ln(x² / (x + 1)). Condensing is the expansion steps run in reverse.

Worked Example — Solving an Exponential Equation

Solve 5·2^x = 320. First isolate the exponential: 2^x = 64. Recognize 64 = 2⁶, or take log base 2 of both sides: x = log₂(64) = 6. When the result isn't a whole power — say 2^x = 50 — use change of base: x = ln(50) / ln(2) ≈ 5.644.

Worked Example — Solving a Logarithmic Equation (and Checking for Extraneous Solutions)

Solve log₃(x) + log₃(x − 2) = 1. Condense with the product property: log₃(x(x − 2)) = 1. Rewrite in exponential form: x(x − 2) = 3¹ = 3, so x² − 2x − 3 = 0 and (x − 3)(x + 1) = 0. The candidates are x = 3 and x = −1. Check each in the ORIGINAL equation: x = 3 works; x = −1 makes log₃(−1) undefined, so it is extraneous. The only solution is x = 3.

Worked Example — Solving an Exponential Inequality

Solve 3^x > 40. Take the natural log of both sides. Because ln is an increasing function, the inequality direction is preserved: x·ln(3) > ln(40), so x > ln(40) / ln(3) ≈ 3.358.

Tip

Every solution of a logarithmic equation must be checked in the original equation, not just the condensed one. Condensing with the product or quotient property can introduce candidate values that make one of the original logarithms undefined.

Common Mistakes

Writing log(M + N) as log(M) + log(N).
The product property applies to log(M·N), not log(M + N). A logarithm of a sum has no simpler equivalent.
Writing log(M) / log(N) as log(M − N) or as log(M/N).
log(M/N) = log(M) − log(N). A quotient of two separate logarithms, log(M)/log(N), is the change-of-base form of log_N(M) and does not collapse to a single logarithm.
Accepting every algebraic solution of a logarithmic equation.
Any candidate that makes an argument of an original logarithm zero or negative is extraneous and must be discarded.

Key Takeaways

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 — each mirrors an exponent law.
Change of base, log_b x = ln x / ln b, rewrites any logarithm for evaluation or for comparing logarithms of different bases.
Solve an exponential equation by isolating the exponential and taking a logarithm; solve a logarithmic equation by condensing to one logarithm, converting to exponential form, then checking for extraneous solutions.
Applying a logarithm to both sides of an inequality preserves its direction, because logarithmic functions are increasing.

Summary

The properties of logarithms rewrite products, quotients, and powers as sums, differences, and coefficients — which is exactly what makes exponential and logarithmic equations solvable exactly rather than only graphically. This closes Unit 2. The next unit turns to a different kind of structure: quantities that rise and fall on a repeating cycle, modeled with trigonometric and polar functions.

Track your progress

← Logarithms as Inverses Back to Unit 2: Exponential & Logarithmic Functions