Logarithm Product Rule

Variables

• b: positive base not equal to one
• x, y: positive arguments

How to use this formula

Converts a logarithm of a product into a sum of logarithms.

Important notes

• The real-valued rule requires positive x and y.
• Do not apply it to log(x+y).

Quick example

log₁₀(100×10)=2+1=3.

Applicability, worked calculation, and verification

Domain and applicability

Apply Logarithm Product Rule only when the listed variables are defined, denominators are nonzero, and the problem satisfies the assumptions stated on this page.

Assumptions and domain checks

• The real-valued rule requires positive x and y.
• Logarithm arguments must be positive in the real domain, and the base must be stated when it is not implied.
• For the Logarithm Product Rule, all variables must lie in the stated domain, and every denominator or inverse operation must be defined.

Do not use this formula when

• Do not use Logarithm Product Rule when its variable definitions, domain restrictions, or structural assumptions differ from the problem.

Boundary and special cases

• For Logarithm Product Rule, check zero, negative, and extreme input values before relying on the result.
• When using Logarithm Product Rule, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values.

Equivalent and alternative forms

• Keep the canonical LaTeX form \log_b(xy)=\log_b x+\log_b y for copying; rearrange only after preserving equivalence and domain restrictions.

Worked example

log₁₀(100×10)=log₁₀100+log₁₀10=2+1=3.

1. Confirm that both factors are positive.
2. Apply the product rule.
3. Evaluate the two simple logarithms and add.

Independent verification

The product is 1000 and log₁₀1000=3.

Common mistakes

• Before substituting values into Logarithm Product Rule, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.
• For the Logarithm Product Rule, check the final value in the original equation; algebraic rearrangement can introduce or lose solutions when domains are restricted.

Review and verification

Last reviewed: 2026-07-26

Review method: Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Logarithm Product Rule.

Verified against: OpenStax Algebra and Trigonometry

Automated quality check: Kept noindex until the missing evidence is supplied.

Formula references

• Algebra and Trigonometry 2eOpenStax, Rice University — Reviewed algebra, functions, sequences, trigonometry, and analytic geometry definitions and examples.

Frequently asked questions