How to Solve Logarithmic Equations

Log equations look intimidating, but the strategy is consistent: isolate the log, rewrite as an exponential, then check the domain.

🧠 Algebra⏱️ ~18 min read📊 Advanced

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What You’ll Learn

✓How to rewrite logs in exponential form
✓When to combine logs using product/quotient/power rules
✓How to handle equations with multiple logs
✓How to check domain restrictions and avoid extraneous solutions

The Core Idea (Log ↔ Exponential)

A logarithm answers the question: “What exponent gives this result?” The key equivalence is:

Log form

log_b(A) = C

Exponential form

b^C = A

For natural logs, the base is e, so ln(A) = C becomes e^C = A.

Important: log arguments must be positive. That means if you see log(x − 3), you must have x − 3 > 0 (so x > 3). This is why checking solutions is not optional in logarithmic equations.

How to Solve Logarithmic Equations (Step-by-Step)

Step 1: Write down domain restrictions

Every log argument must be positive. If you have multiple logs, you’ll have multiple restrictions. You don’t always need to solve them immediately, but you should keep them in mind so you can reject extraneous answers later.

Step 2: Isolate the log expression(s)

Use normal algebra (add/subtract/multiply/divide) to get a single log on one side whenever possible. If logs are already alone, you’re ready to convert.

Step 3: Combine logs (when helpful)

When you have two logs on the same side, log rules can turn them into one log:

• Product: log(A) + log(B) = log(AB)
• Quotient: log(A) − log(B) = log(A/B)
• Power: k·log(A) = log(A^k)

These rules help you reach the classic pattern log(…) = number, which you can then rewrite exponentially.

Step 4: Rewrite as an exponential equation

Convert log_b(A) = C into b^C = A. This is where logarithms stop being “mysterious” and turn into an algebra problem.

Step 5: Solve, then check

Solve the resulting equation for the variable. Then substitute your solution(s) back into the original logarithmic equation and verify all arguments are positive.

If you get a value that makes any log argument 0 or negative, that value is not a valid solution.

Worked Example 1 (Single Log)

Solve: log_2(x − 3) = 5

Domain restriction: x − 3 > 0 so x > 3.
Rewrite in exponential form: 2^5 = x − 3.
Compute: 32 = x − 3 so x = 35.
Check: x − 3 = 32 > 0 so the solution is valid.

Worked Example 2 (Two Logs)

Solve: ln(x) + ln(x − 1) = ln(12)

Domain restrictions: x > 0 and x − 1 > 0, so x > 1.
Combine logs using the product rule: ln(x(x − 1)) = ln(12).
If ln(A) = ln(B), then A = B (for positive A and B): x(x − 1) = 12.
Solve the quadratic: x^2 − x − 12 = 0 which factors to (x − 4)(x + 3) = 0. So x = 4 or x = −3.
Check against the domain restriction x > 1: only x = 4 is valid.

Notice how a log equation can turn into a quadratic. If you need a refresher on solving quadratics, see how to solve a quadratic equation step by step.

Common Mistakes (and How to Avoid Them)

Skipping the domain check

Log arguments must be positive. Many “wrong answers” are actually extraneous solutions that fail this requirement.

Misusing log rules

There is no rule like log(A + B) = log(A) + log(B). Only products/quotients/powers are safe.

Forgetting parentheses

When rewriting to exponentials, be careful with negatives and fractions in the exponent.

Dropping the base

log usually means base 10, ln means base e, and log_b means base b. The base matters in the exponential form.

If you want a quick refresher on the log rules used above, the OpenStax section on logarithmic functions is a solid reference.

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