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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

Why it matters: Exponential and logarithmic models describe growth, decay, sound, pH, finance, and scientific scales.

Worked example

Problem. Apply the product rule to expand log(3 times 4).

  1. The product rule turns a product inside the log into a sum outside.
  2. log(3 times 4) = log(3) + log(4).
  3. Multiplication inside becomes addition outside.

Answer: log(3) + log(4)

Practice problems

1. Apply the product rule to expand log(3 times 4).

Choices: log(3) + log(4) · log(3) - log(4) · log(3) log(4) · log(3 + 4)

Show solution
  1. The product rule turns a product inside the log into a sum outside.
  2. log(3 times 4) = log(3) + log(4).
  3. Multiplication inside becomes addition outside.

Answer: log(3) + log(4)

2. Apply the quotient rule to expand log(4/5).

Choices: log(4) - log(5) · log(4) + log(5) · log(4)/log(5) · log(4 - 5)

Show solution
  1. The quotient rule turns a quotient inside the log into a difference.
  2. log(4/5) = log(4) - log(5).
  3. Division inside becomes subtraction outside, not a quotient of logs.

Answer: log(4) - log(5)

3. Apply the power rule to rewrite log(x^2).

Choices: 2log(x) · log(2x) · log(x) + 2 · xlog(2)

Show solution
  1. Core Practice: First identify exactly what the question is asking: Apply the power rule to rewrite log(x^2).
  2. For logarithms, rewrite the statement as an exponent question so the base, exponent, and result are clear.
  3. The power rule moves the exponent in front as a coefficient.
  4. log(x^2) = 2log(x).
  5. The exponent becomes a multiplier.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: 2log(x)

4. Which logarithm 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
  1. Valid rules act on products, quotients, and powers inside the log.
  2. log(ab) = log(a) + log(b) is the product rule.
  3. There is no rule for a sum inside the log.

Answer: log(ab) = log(a) + log(b)

5. Expand log(xy) using a log rule.

Choices: log(x) + log(y) · log(x) - log(y) · log(x)log(y) · log(x + y)

Show solution
  1. The argument xy is a product.
  2. Apply the product rule: log(xy) = log(x) + log(y).
  3. A product inside becomes a sum outside.

Answer: log(x) + log(y)

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