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Logarithm Basics

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

A logarithm answers an exponent question. The statement log base b of x = y means b raised to the y power equals x. Logarithms matter because they undo exponential functions, which allows us to solve for unknown exponents. When practicing, rewrite each logarithm as an exponential equation before solving. Pay attention to the base, the output, and the argument of the logarithm. A common mistake is treating log notation like ordinary multiplication or forgetting that the base must be positive and not equal to 1.

What you'll learn

Why it matters: Earthquake magnitudes (Richter), sound levels (decibels), pH, and stellar brightness all use logarithms. A log answers 'what exponent makes this base reach that value?' — and the answer compresses huge ranges into something readable.

Worked example

Problem. Evaluate log base 2 of 8.

  1. Ask: 2 to what power equals 8?
  2. 2^3 = 8.
  3. So log base 2 of 8 is 3.

Answer: 3

Practice problems

1. Evaluate log base 10 of 100.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Evaluate log base 10 of 100.
  2. For logarithms, rewrite the statement as an exponent question so the base, exponent, and result are clear.
  3. 10^2 = 100.
  4. Check the result by substituting or estimating: the response should match 2 and make sense in the original problem.

Answer: 2

2. 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. 3^3 = 27.
  4. Check the result by substituting or estimating: the response should match 3 and make sense in the original problem.

Answer: 3

3. Evaluate log base 5 of 25.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Evaluate log base 5 of 25.
  2. For logarithms, rewrite the statement as an exponent question so the base, exponent, and result are clear.
  3. 5^2 = 25.
  4. Check the result by substituting or estimating: the response should match 2 and make sense in the original problem.

Answer: 2

4. log base 2 of 16 = 4 means...

Choices: 2^4 = 16 · 4^2 = 16 · 16^2 = 4 · 2 + 4 = 16

Show solution
  1. Core Practice: First identify exactly what the question is asking: log base 2 of 16 = 4 means...
  2. For logarithms, rewrite the statement as an exponent question so the base, exponent, and result are clear.
  3. Rewrite the logarithm as an exponential.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: 2^4 = 16

5. Evaluate log base 4 of 1.

Show solution
  1. Challenge: First identify exactly what the question is asking: Evaluate log base 4 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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