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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 9.

  1. Worked Example: First identify exactly what the question is asking: Evaluate log base 3 of 9.
  2. For logarithms, rewrite the statement as an exponent question so the base, exponent, and result are clear.
  3. A logarithm asks: 3 to what power gives 9?
  4. 3^2 = 9.
  5. So log base 3 of 9 = 2.
  6. Check the result by substituting or estimating: the response should match 2 and make sense in the original problem.

Answer: 2

Practice problems

1. Evaluate log base 3 of 9.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Evaluate log base 3 of 9.
  2. For logarithms, rewrite the statement as an exponent question so the base, exponent, and result are clear.
  3. A logarithm asks: 3 to what power gives 9?
  4. 3^2 = 9.
  5. So log base 3 of 9 = 2.
  6. Check the result by substituting or estimating: the response should match 2 and make sense in the original problem.

Answer: 2

2. Rewrite log base 2 of 32 = 5 in exponential form.

Choices: 2^5 = 32 · 2 × 5 = 32 · 2^32 = 5 · 32^5 = 2

Show solution
  1. Warm-up: First identify exactly what the question is asking: Rewrite log base 2 of 32 = 5 in exponential form.
  2. For logarithms, rewrite the statement as an exponent question so the base, exponent, and result are clear.
  3. A logarithm names an exponent: the answer 5 is the power.
  4. The base 2 is raised to that power.
  5. So 2^5 = 32.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: 2^5 = 32

3. What is the value of log base 3 of 1?

Show solution
  1. Core Practice: First identify exactly what the question is asking: What is the value of log base 3 of 1?
  2. For logarithms, rewrite the statement as an exponent question so the base, exponent, and result are clear.
  3. Ask: 3 to what power gives 1?
  4. Any nonzero base to the 0 power is 1, so 3^0 = 1.
  5. So log base 3 of 1 = 0.
  6. Check the result by substituting or estimating: the response should match 0 and make sense in the original problem.

Answer: 0

4. A log of its own base: find log base 2 of 2.

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

Answer: 1

5. Simplify 3^(log base 3 of 81).

Choices: 81 · 3 · 4 · 12

Show solution
  1. The exponential and the logarithm with base 3 are inverses.
  2. Raising 3 to the log base 3 of a value returns that value.
  3. So 3^(log base 3 of 81) = 81.

Answer: 81

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