Logarithmic Equations and Applications
A free College Algebra lesson from the “Logarithms” unit, with a worked example and practice problems including step-by-step solutions.
Logarithms solve exponential equations and compress multiplicative relationships. Product, quotient, and power properties come from exponent rules.
What you'll learn
- Use log properties
- Solve simple exponential equations
- Interpret logarithmic models
Worked example
Problem. Solve 2^x = 64.
- Rewrite 64 as 2^6.
- 2^x = 2^6.
- x = 6.
Answer: 6
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. Solve 3^x = 27.
Show solution
- 27 = 3^3.
Answer: 3
2. log_b(xy) equals...
Choices: log_b(x) + log_b(y) · log_b(x) - log_b(y) · log_b(x/y) · log_b(x + y)
Show solution
- Product property.
Answer: log_b(x) + log_b(y)
3. log_b(x^4) equals...
Choices: 4log_b(x) · log_b(4x) · log_b(x) + 4 · xlog_b(4)
Show solution
- Power property.
Answer: 4log_b(x)
4. Solve 10^x = 10000.
Show solution
- 10000 = 10^4.
Answer: 4
5. log_b(x) + log_b(4) equals...
Choices: log_b(4x) · log_b(x/4) · log_b(x + 4) · 4log_b(x)
Show solution
- Add logs with the same base by multiplying inputs.
Answer: log_b(4x)
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