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
Practice problems
1. Solve 3^x = 27.
Show solution
- Warm-up: First identify exactly what the question is asking: Solve 3^x = 27.
- Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
- 27 = 3^3.
- Check the result by substituting or estimating: the response should match 3 and make sense in the original problem.
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
- Core Practice: First identify exactly what the question is asking: log_b(xy) equals...
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- Product property.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
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
- Challenge: First identify exactly what the question is asking: log_b(x^4) equals...
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- Power property.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: 4log_b(x)
4. Solve 10^x = 10000.
Show solution
- Exponential Equations: First identify exactly what the question is asking: Solve 10^x = 10000.
- For exponential situations, identify the starting value and the repeated multiplier before calculating.
- 10000 = 10^4.
- Check the result by substituting or estimating: the response should match 4 and make sense in the original problem.
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
- Log Properties: First identify exactly what the question is asking: log_b(x) + log_b(4) equals...
- For logarithms, rewrite the statement as an exponent question so the base, exponent, and result are clear.
- Add logs with the same base by multiplying inputs.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: log_b(4x)
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