Logarithm Properties
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.
Logarithm properties come from exponent rules. Products become sums, quotients become differences, and powers move to the front as coefficients.
What you'll learn
- Use product, quotient, and power properties
- Expand logarithms
- Condense logarithms
Worked example
Problem. Expand log base b of (xy).
- A product inside a log becomes a sum of logs.
- Keep the same base.
- So log_b(xy) = log_b(x) + log_b(y).
Answer: log_b(x) + log_b(y)
Lesson preview: this page shows 5 practice items. The interactive activity contains 28 practice items and 15 quiz items (multiple choice and typed response). Counts include repeated prompts and variants.
Practice problems
1. log_b(xy) equals...
Choices: log_b(x) + log_b(y) · log_b(x) - log_b(y) · log_b(x)/log_b(y) · log_b(x + y)
Show solution
- Product property.
Answer: log_b(x) + log_b(y)
2. log_b(x/y) equals...
Choices: log_b(x) - log_b(y) · log_b(x) + log_b(y) · log_b(xy) · log_b(y) - log_b(x)
Show solution
- Quotient property.
Answer: log_b(x) - log_b(y)
3. log_b(x^5) equals...
Choices: 5log_b(x) · log_b(5x) · log_b(x) + 5 · xlog_b(5)
Show solution
- Power property.
Answer: 5log_b(x)
4. log_b(3) + log_b(7) condenses to...
Choices: log_b(21) · log_b(10) · log_b(4) · log_b(3/7)
Show solution
- Sum of logs becomes log of product.
Answer: log_b(21)
5. Evaluate log base 10 of 1000.
Show solution
- A logarithm asks for the exponent.
- 10^3 = 1000.
- So log base 10 of 1000 is 3.
Answer: 3
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