Exponential & Logarithmic Functions

Solving Exponential & Logarithmic Equations

Solving equations by taking a logarithm of both sides or rewriting in exponential form.

Advanced20 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

Properties of Logarithms

Freeing a Variable from an Exponent

2^x = 8 solves easily, since both sides can be rewritten with base 2. Before reading on, think about 3^x = 20 — no whole-number exponent gives exactly 20, so matching bases won't work here. What tool from the last two lessons could still isolate x?

Taking the logarithm of both sides is exactly the move that works generally, since a logarithm undoes exponentiation. Once a log is applied to both sides, the power property pulls the exponent out front as a plain multiplying coefficient — turning an equation with x buried in an exponent into an ordinary equation solvable with algebra.

Worked Example — Solving by Matching Bases

Solve 4^(x+1) = 64. Rewrite 64 as a power of 4: 64 = 4³. Since the bases match, the exponents must be equal: x + 1 = 3, so x = 2.

Worked Example — Solving by Taking a Logarithm of Both Sides

Solve 3^x = 20. Take the log of both sides: log(3^x) = log(20). Apply the power property: x · log(3) = log(20). Isolate x: x = log(20)/log(3) ≈ 1.301/0.477 ≈ 2.727.

Worked Example — Solving a Logarithmic Equation

Solve log_2(x + 3) = 4. Rewrite in exponential form: 2⁴ = x + 3, so 16 = x + 3, giving x = 13. Check: log_2(13 + 3) = log_2(16) = 4. Valid, and the argument x + 3 = 16 is positive, so it's within the logarithm's domain.

Tip

For any logarithmic equation, check the solution against the logarithm's domain before accepting it — a candidate solution that makes an argument inside a logarithm zero or negative is extraneous, the same domain-restriction danger that made radical equations worth double-checking.

Common Mistakes

Dividing log(20) by log(3) as if it simplified to log(20/3), confusing a quotient of logarithms with the logarithm of a quotient.
log(20)/log(3) is a division of two separate logarithm values — it is not the same as log(20/3), the quotient property's actual meaning; these are two genuinely different expressions.
Accepting a solution to a logarithmic equation without checking that every logarithm's argument stays positive.
Substitute the solution back into the original equation and confirm every argument inside a logarithm is positive — a solution making an argument zero or negative must be rejected as extraneous.

Key Takeaways

When bases can be matched, exponents can be set equal directly.
When bases can't be matched, taking the logarithm of both sides and applying the power property isolates the variable.
Every logarithmic equation's solution must be checked to ensure it keeps every logarithm's argument positive.

Summary

Logarithms turn otherwise unsolvable exponential equations into ordinary algebra. The next unit turns to rational functions, where division itself — not just exponents — shapes a function's graph.

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