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Logarithms as Inverses

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

A logarithm answers an exponent question: base to what power gives this value? 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. Evaluate log base 3 of 27.

  1. Worked Example: First identify exactly what the question is asking: Evaluate log base 3 of 27.
  2. For logarithms, rewrite the statement as an exponent question so the base, exponent, and result are clear.
  3. Ask: 3 to what power equals 27?
  4. 3^3 = 27.
  5. The log value is 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. Evaluate log base 3 of 27.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Evaluate log base 3 of 27.
  2. For logarithms, rewrite the statement as an exponent question so the base, exponent, and result are clear.
  3. Ask: 3 to what power equals 27?
  4. 3^3 = 27.
  5. The log value is 3.
  6. Check the result by substituting or estimating: the response should match 3 and make sense in the original problem.

Answer: 3

2. Evaluate log base 4 of 256.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Evaluate log base 4 of 256.
  2. For logarithms, rewrite the statement as an exponent question so the base, exponent, and result are clear.
  3. Ask: 4 to what power equals 256?
  4. 4^4 = 256.
  5. The log value is 4.
  6. Check the result by substituting or estimating: the response should match 4 and make sense in the original problem.

Answer: 4

3. Evaluate log base 2 of 32.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Evaluate log base 2 of 32.
  2. For logarithms, rewrite the statement as an exponent question so the base, exponent, and result are clear.
  3. Ask: 2 to what power equals 32?
  4. 2^5 = 32.
  5. The log value is 5.
  6. Check the result by substituting or estimating: the response should match 5 and make sense in the original problem.

Answer: 5

4. Which exponential form matches log base 3 of 9 = 2?

Choices: 3^2 = 9 · 9^3 = 2 · 3^9 = 2 · 2^3 = 9

Show solution
  1. Core Practice: First identify exactly what the question is asking: Which exponential form matches log base 3 of 9 = 2?
  2. For logarithms, rewrite the statement as an exponent question so the base, exponent, and result are clear.
  3. A logarithm is an exponent question.
  4. The base stays 3.
  5. The exponent is 2 and the result is 9.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: 3^2 = 9

5. Which exponential form matches log base 4 of 64 = 3?

Choices: 4^3 = 64 · 64^4 = 3 · 4^64 = 3 · 3^4 = 64

Show solution
  1. Core Practice: First identify exactly what the question is asking: Which exponential form matches log base 4 of 64 = 3?
  2. For logarithms, rewrite the statement as an exponent question so the base, exponent, and result are clear.
  3. A logarithm is an exponent question.
  4. The base stays 4.
  5. The exponent is 3 and the result is 64.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: 4^3 = 64

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