Solving Exponential Equations Examples in Math

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

Using logarithms to solve equations where the unknown is in the exponent, such as a^x = b.

When the variable is trapped in an exponent, logarithms free it. Taking \log of both sides brings the exponent down to ground level where you can solve for it using algebra.

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 strategy is always: isolate the exponential term, take the logarithm of both sides, then use log properties to solve for the variable.

Common stuck point: When the equation has exponentials on BOTH sides (like 2^x = 3^{x-1}), take \ln of both sides and collect terms with x on one side.

Sense of Study hint: Isolate the exponential term on one side first, then take ln of both sides. Use the power rule to bring the exponent down and solve for x.

Worked Examples

Example 1

easy
Solve 3^x = 81.

Solution

  1. 1
    Recognize that 81 is a power of 3: 81 = 3^4.
  2. 2
    So the equation becomes 3^x = 3^4.
  3. 3
    Since the bases are equal, the exponents must be equal: x = 4.

Answer

x = 4
When both sides of an exponential equation can be written with the same base, set the exponents equal. This is the simplest method and avoids logarithms entirely. Always check if the number is a recognizable power of the base.

Example 2

medium
Solve 5^{2x-1} = 12.

Practice Problems

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

Example 1

medium
Solve 2^{x+3} = 7^{x-1}.

Example 2

hard
Solve e^{2x} - 5e^x + 6 = 0.
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Background Knowledge

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

exponential functionlogarithmlogarithm properties