Solving Logarithmic Equations Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Solving Logarithmic Equations.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Solving equations containing logarithms by converting to exponential form or using log properties to combine and simplify.

If logarithms trap the variable inside a \log, converting to exponential form releases it. The key insight is that \log_b(\text{stuff}) = c means b^c = \text{stuff}—just rewrite and solve.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: The main strategies are: (1) convert to exponential form, (2) combine multiple logs into one using log properties, then convert, and (3) always check for extraneous solutions since log arguments must be positive.

Common stuck point: You MUST check your solutions. Logarithms require positive arguments, so a solution that makes any \log argument zero or negative is extraneous and must be rejected.

Sense of Study hint: Convert the log equation to exponential form: log_b(stuff) = c becomes b^c = stuff. Then solve the resulting equation and check the answer.

Worked Examples

Example 1

easy
Solve \log_2(x) = 5.

Solution

  1. 1
    Convert from logarithmic to exponential form: \log_2(x) = 5 means 2^5 = x.
  2. 2
    Calculate: x = 2^5 = 32.
  3. 3
    Check: \log_2(32) = \log_2(2^5) = 5. ✓

Answer

x = 32
The fundamental relationship between logarithms and exponents is: \log_b(a) = c if and only if b^c = a. Converting to exponential form is the most direct way to solve simple logarithmic equations.

Example 2

medium
Solve \log(x+3) + \log(x-1) = \log(5).

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

medium
Solve \ln(x^2 - 4) = \ln(3x).

Example 2

hard
Solve \log_3(x+6) - \log_3(x-2) = 2.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

logarithmlogarithm properties