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Logarithms

A free College Algebra lesson from the “Logarithms” unit, with a worked example and practice problems including step-by-step solutions.

A logarithm is the inverse of an exponential expression. The statement log base b of x = y means b^y = x. Logarithms matter because they let us solve equations where the unknown is in the exponent and because they compress multiplicative growth into additive scale. When practicing, rewrite logs in exponential form, identify the base, and check that the argument is positive. A common mistake is ignoring the base or treating logarithms as if they distribute over addition; log(a + b) is not the same as log(a) + log(b).

What you'll learn

Why it matters: Sound levels, pH, earthquake magnitude, and growth-time questions all use logarithms to answer an exponent question. Rewriting a logarithm as an exponential statement makes the inverse relationship concrete.

Worked example

Problem. Evaluate log base 2 of 32.

  1. Ask: 2 to what power equals 32?
  2. 2^5 = 32.
  3. The logarithm is 5.

Answer: 5

Practice problems

1. Evaluate log base 10 of 1000.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Evaluate log base 10 of 1000.
  2. For logarithms, rewrite the statement as an exponent question so the base, exponent, and result are clear.
  3. 10^3 = 1000.
  4. 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 3 of 81.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Evaluate log base 3 of 81.
  2. For logarithms, rewrite the statement as an exponent question so the base, exponent, and result are clear.
  3. 3^4 = 81.
  4. Check the result by substituting or estimating: the response should match 4 and make sense in the original problem.

Answer: 4

3. log base 5 of 25 = 2 means...

Choices: 5^2 = 25 · 25^2 = 5 · 2^5 = 25 · 5 + 2 = 25

Show solution
  1. Challenge: First identify exactly what the question is asking: log base 5 of 25 = 2 means...
  2. For logarithms, rewrite the statement as an exponent question so the base, exponent, and result are clear.
  3. Convert to exponential form.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: 5^2 = 25

4. Evaluate log base 2 of 8.

Show solution
  1. Evaluation: First identify exactly what the question is asking: Evaluate log base 2 of 8.
  2. For logarithms, rewrite the statement as an exponent question so the base, exponent, and result are clear.
  3. 2^3 = 8.
  4. Check the result by substituting or estimating: the response should match 3 and make sense in the original problem.

Answer: 3

5. Evaluate log base 9 of 1.

Show solution
  1. Evaluation: First identify exactly what the question is asking: Evaluate log base 9 of 1.
  2. For logarithms, rewrite the statement as an exponent question so the base, exponent, and result are clear.
  3. Any nonzero base to the 0 power is 1.
  4. Check the result by substituting or estimating: the response should match 0 and make sense in the original problem.

Answer: 0

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