Algebra 2 / Precalculus · Step-by-step guide

How to Solve Logarithmic Equations

Solve logarithmic equations by rewriting exponential form, combining logs, and rejecting extraneous solutions.

By Tyler Blovat··9 min read
The short answer

Start with the central idea

Isolate or combine the logarithm, rewrite log⁡b(A)=c\displaystyle \log_b(A)=c as A=bc\displaystyle A=b^c, solve the resulting equation, and check every candidate in the original because logarithm arguments must stay positive.

01

How it works

Build the method from meaning before memorizing the moves.

These are the decisions that make the procedure reliable. Read them once, then look for each idea in the worked examples below.

1

Use one logarithm

Combine same-base logs when possible before converting to exponential form.

2

Match logs carefully

If log⁡bM=log⁡bN\displaystyle \log_bM=\log_bN, then M = N, provided both arguments are positive.

3

Check the domain

Algebra can produce candidates that make an original logarithm zero or negative; those must be rejected.

02

See the structure

A picture makes the relationships easier to remember.

The curve passes through its initial value and rises increasingly quickly for a growth factor greater than 1.
03

Worked examples

Follow the reason for each line, then try to reproduce it without looking.

Example 1Convert forms

Solve log⁡2(x−1)=4\displaystyle \log_2(x-1)=4.

  1. Rewrite as x−1=24\displaystyle x-1=2^4.
  2. Then x−1=16\displaystyle x-1=16.
  3. The argument 16 is positive.
Answerx=17\displaystyle x=17
Example 2Combine logs

Solve ln⁡x+ln⁡(x−3)=ln⁡4\displaystyle \ln x+\ln(x-3)=\ln4.

  1. Combine: ln⁡(x(x−3))=ln⁡4\displaystyle \ln(x(x-3))=\ln4.
  2. Solve x2−3x−4=0\displaystyle x^2-3x-4=0.
  3. Candidates are 4 and −1; reject −1.
Answerx=4\displaystyle x=4
Example 3Same base

Solve log⁡5(2x+1)=log⁡5(7)\displaystyle \log_5(2x+1)=\log_5(7).

  1. Set arguments equal.
  2. 2x+1=7\displaystyle 2x+1=7.
  3. Check the original argument is positive.
Answerx=3\displaystyle x=3
04

Common mistakes—and how to avoid them

Accuracy improves fastest when you know what to check.

  • Exponentiating before isolating the logarithm.

  • Keeping solutions that make any log argument nonpositive.

  • Using log rules across addition inside an argument.

05

Turn the explanation into a skill

Reading creates recognition. Independent practice creates recall.

Free printable worksheet · Answer key included

Logarithms

Practice log properties and equations in a connected set.

Preview the worksheet

A strong practice loop: solve one example with the guide open, solve a similar problem without it, explain the method aloud, and return the next day for a short mixed review.

06

Practice problems with worked solutions

Solve each problem on paper, use the hint only if needed, then compare every step.

1

Solve log⁡2x=5\displaystyle \log_2 x=5.

Show hint

Rewrite in exponential form.

Show worked solution
  1. x=25\displaystyle x=2^5.
  2. x=32>0\displaystyle x=32>0, so it is valid.

Answer: x=32\displaystyle x=32

2

Solve ln⁡(x−1)=2\displaystyle \ln(x-1)=2.

Show hint

Exponentiate both sides.

Show worked solution
  1. x−1=e2\displaystyle x-1=e^2.
  2. Add 1.
  3. The argument is positive.

Answer: x=1+e2\displaystyle x=1+e^2

3

Solve log⁡x+log⁡(x−9)=1\displaystyle \log x+\log(x-9)=1.

Show hint

Combine the logs before rewriting.

Show worked solution
  1. log⁡(x(x−9))=1\displaystyle \log(x(x-9))=1.
  2. x2−9x=10\displaystyle x^2-9x=10.
  3. x=10\displaystyle x=10 or −1, but −1 makes log arguments invalid.

Answer: x=10\displaystyle x=10

4

Solve log⁡3(x+1)−log⁡3(x−1)=1\displaystyle \log_3(x+1)-\log_3(x-1)=1.

Show hint

Use the quotient rule.

Show worked solution
  1. log⁡3((x+1)/(x−1))=1\displaystyle \log_3((x+1)/(x-1))=1.
  2. (x+1)/(x−1)=3\displaystyle (x+1)/(x-1)=3.
  3. x+1=3x−3\displaystyle x+1=3x-3, so x=2\displaystyle x=2.

Answer: x=2\displaystyle x=2

5

Solve 2ln⁡x=ln⁡16\displaystyle 2\ln x=\ln16.

Show hint

Use the power rule.

Show worked solution
  1. ln⁡(x2)=ln⁡16\displaystyle \ln(x^2)=\ln16.
  2. x2=16\displaystyle x^2=16.
  3. A logarithm requires x>0\displaystyle x>0.

Answer: x=4\displaystyle x=4

07

Frequently asked questions

Quick answers before you move on.

What should I remember first?

Isolate or combine the logarithm, rewrite log⁡b(A)=c\displaystyle \log_b(A)=c as A=bc\displaystyle A=b^c, solve the resulting equation, and check every candidate in the original because logarithm arguments must stay positive.

How do I know whether I understand this topic?

You should be able to name the method, explain why each step is valid, complete a new example without copying, and check whether your answer is reasonable.

What should I do if I keep making the same mistake?

Write the error as a specific checkpoint—for example, “I will identify the hypotenuse before substituting.” Then use that checkpoint on three short problems rather than repeating a full page without feedback.

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