Product, Quotient, and Power Rules

Algebra Foundation

Algebra 2

Explore advanced functions, complex numbers, polynomials, rational and radical expressions, exponentials, logarithms, conics, sequences, introductory trigonometry, probability, and statistics.

Learn with original explanations and worked examples. OpenStax Intermediate Algebra 2e was consulted for topic coverage.

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Algebra 2 · Unit 7 · Topic 7.2

Product, Quotient, and Power Rules

For positive M and N and a valid base b, log_b(MN) = log_b(M) + log_b(N), log_b(M/N) = log_b(M) − log_b(N), and log_b(Mʳ) = r log_b(M).

Definition and explanation

For positive M and N and a valid base b, log_b(MN) = log_b(M) + log_b(N), log_b(M/N) = log_b(M) − log_b(N), and log_b(Mʳ) = r log_b(M). Domain assumptions are part of these identities.

Worked example

For x > 0, expand ln(5x³) as ln(5) + 3ln(x). If x = e, the expression becomes ln(5) + 3, matching ln(5e³).

Common mistake

There is no corresponding rule ln(M + N) = ln(M) + ln(N); multiplication and addition are different operations.

Key takeaway

For positive M and N and a valid base b, log_b(MN) = log_b(M) + log_b(N), log_b(M/N) = log_b(M) − log_b(N), and log_b(Mʳ) = r log_b(M).

Explain each step of the worked example, then check the units, signs, and conditions that apply.

Further reading

Topic coverage was checked against the openly available OpenStax Intermediate Algebra 2e, Algebra and Trigonometry 2e, and Introductory Statistics 2e textbooks. The explanations and examples on this site are independently written; textbook prose, exercises, and artwork are not reproduced.