Logarithm Rules
A free Precalculus lesson from the “Exponential and Logarithmic Functions” unit, with a worked example and practice problems including step-by-step solutions.
Log rules come from exponent rules. They apply to products, quotients, and powers, not to sums inside the log. This lesson is part of Precalculus: Advanced Functions, so the emphasis is on interpreting behavior, choosing the right representation, and explaining the result clearly rather than memorizing isolated algebra moves.
What you'll learn
- Use product, quotient, and power rules without inventing false rules
- Use logarithm rules in symbolic and graph-based problems
- Check common mistakes before finalizing an answer
Worked example
Problem. Which rule is valid?
- The product rule changes multiplication inside into addition outside.
- It does not apply to addition inside.
- Powers become coefficients.
Answer: log(ab) = log(a) + log(b)
Practice problems
1. Which rule is valid?
Choices: log(ab) = log(a) + log(b) · log(a + b) = log(a) + log(b) · log(a/b) = log(a)log(b) · log(a^k) = log(a) + k
Show solution
- The product rule changes multiplication inside into addition outside.
- It does not apply to addition inside.
- Powers become coefficients.
Answer: log(ab) = log(a) + log(b)
2. Which rule is valid? (variation 2)
Choices: log(ab) = log(a) + log(b) · log(a + b) = log(a) + log(b) · log(a/b) = log(a)log(b) · log(a^k) = log(a) + k
Show solution
- The product rule changes multiplication inside into addition outside.
- It does not apply to addition inside.
- Powers become coefficients.
Answer: log(ab) = log(a) + log(b)
3. Use the power rule to rewrite log(x^5).
Choices: 5log(x) · log(5x) · log(x) + 5 · xlog(5)
Show solution
- Core Practice: First identify exactly what the question is asking: Use the power rule to rewrite log(x^5).
- For logarithms, rewrite the statement as an exponent question so the base, exponent, and result are clear.
- The exponent becomes a coefficient.
- Keep the log of the base expression.
- So log(x^5) = 5log(x).
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: 5log(x)
4. Use the power rule to rewrite log(x^5). (variation 2)
Choices: 5log(x) · log(5x) · log(x) + 5 · xlog(5)
Show solution
- Core Practice: First identify exactly what the question is asking: Use the power rule to rewrite log(x^5).
- For logarithms, rewrite the statement as an exponent question so the base, exponent, and result are clear.
- The exponent becomes a coefficient.
- Keep the log of the base expression.
- So log(x^5) = 5log(x).
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: 5log(x)
5. A common log-rule mistake is:
Choices: splitting log(a + b) into log(a) + log(b) · using log(ab) = log(a) + log(b) · moving an exponent down as a coefficient · rewriting a quotient as a difference
Show solution
- Core Practice: First identify exactly what the question is asking: A common log-rule mistake is:
- For logarithms, rewrite the statement as an exponent question so the base, exponent, and result are clear.
- There is no sum rule for logs.
- Product, quotient, and power rules are valid.
- Addition inside a log must stay together.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: splitting log(a + b) into log(a) + log(b)
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