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
- Interpret logarithms
- Convert between exponential and logarithmic form
- Evaluate common logarithms
Worked example
Problem. Evaluate log base 2 of 32.
- Ask: 2 to what power equals 32?
- 2^5 = 32.
- The logarithm is 5.
Answer: 5
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 1000.
Show solution
- 10^3 = 1000.
Answer: 3
2. Evaluate log base 3 of 81.
Show solution
- 3^4 = 81.
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
- Convert to exponential form.
Answer: 5^2 = 25
4. Evaluate log base 2 of 8.
Show solution
- 2^3 = 8.
Answer: 3
5. Evaluate log base 9 of 1.
Show solution
- Any nonzero base to the 0 power is 1.
Answer: 0
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