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Solving Exponential and Logarithmic Equations

A free Precalculus lesson from the “Exponential and Logarithmic Functions” unit, with a worked example and practice problems including step-by-step solutions.

Solving exponential and log equations means choosing the inverse form and checking any domain restrictions. This lesson is part of Precalculus: Advanced Functions, so the emphasis is on interpreting behavior, choosing the right representation, and explaining the result clearly rather than memorizing isolated algebra moves.

What you'll learn

Why it matters: Exponential and logarithmic models describe growth, decay, sound, pH, finance, and scientific scales.

Worked example

Problem. Solve 3^x = 27.

  1. Worked Example: First identify exactly what the question is asking: Solve 3^x = 27.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Rewrite 27 as 3^3.
  4. Now the bases match.
  5. So x = 3.
  6. Check the result by substituting or estimating: the response should match 3 and make sense in the original problem.

Answer: 3

Practice problems

1. Solve 3^x = 27.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Solve 3^x = 27.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Rewrite 27 as 3^3.
  4. Now the bases match.
  5. So x = 3.
  6. Check the result by substituting or estimating: the response should match 3 and make sense in the original problem.

Answer: 3

2. Solve 4^x = 256.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Solve 4^x = 256.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Rewrite 256 as 4^4.
  4. Now the bases match.
  5. So x = 4.
  6. Check the result by substituting or estimating: the response should match 4 and make sense in the original problem.

Answer: 4

3. Solve log base 2 of x = 5.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Solve log base 2 of x = 5.
  2. For logarithms, rewrite the statement as an exponent question so the base, exponent, and result are clear.
  3. Rewrite in exponential form.
  4. 2^5 = x.
  5. So x = 32.
  6. Check the result by substituting or estimating: the response should match 32 and make sense in the original problem.

Answer: 32

4. Solve log base 3 of x = 2.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Solve log base 3 of x = 2.
  2. For logarithms, rewrite the statement as an exponent question so the base, exponent, and result are clear.
  3. Rewrite in exponential form.
  4. 3^2 = x.
  5. So x = 9.
  6. Check the result by substituting or estimating: the response should match 9 and make sense in the original problem.

Answer: 9

5. When solving a logarithmic equation, a final check should include:

Choices: that each log input is positive · that every answer is negative · that bases disappear · that no exponents were used

Show solution
  1. Core Practice: First identify exactly what the question is asking: When solving a logarithmic equation, a final check should include:
  2. For logarithms, rewrite the statement as an exponent question so the base, exponent, and result are clear.
  3. Logarithms require positive inputs.
  4. Algebra can produce extraneous answers.
  5. Check the original equation.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: that each log input is positive

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