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
- Interpret logarithms
- Convert between exponential and logarithmic form
- Evaluate simple logarithms
Worked example
Problem. Evaluate log base 2 of 8.
- Ask: 2 to what power equals 8?
- 2^3 = 8.
- So log base 2 of 8 is 3.
Answer: 3
Lesson preview: this page shows 5 practice items. The interactive activity contains 28 practice items and 15 quiz items (typed response and multiple choice). Counts include repeated prompts and variants.
Practice problems
1. Evaluate log base 10 of 100.
Show solution
- 10^2 = 100.
Answer: 2
2. Evaluate log base 3 of 27.
Show solution
- 3^3 = 27.
Answer: 3
3. Evaluate log base 5 of 25.
Show solution
- 5^2 = 25.
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
- Rewrite the logarithm as an exponential.
Answer: 2^4 = 16
5. Evaluate log base 4 of 1.
Show solution
- Any nonzero base to the 0 power is 1.
Answer: 0
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