Logarithm Product Rule
The logarithm of a product equals the sum of logarithms.
Formula
Variables
Example
log₂(8×4) = 3 + 2 = 5
How it works
A logarithm tells you what exponent a base must be raised to in order to produce a number. Since multiplying two powers with the same base adds their exponents, taking the log of a product turns that multiplication into addition of the separate logarithms. This works only when both factors are positive and the base is valid for logarithms.
When to use it
Use this rule when a logarithm contains a product inside the brackets and you want to simplify it into separate terms. It is especially helpful in algebra, calculus, and scientific calculations where products are easier to manage as sums.
Worked example, step by step
Find log_3(9 × 27) log_3(9 × 27) = log_3(9) + log_3(27) Since 9 = 3^2, log_3(9) = 2 Since 27 = 3^3, log_3(27) = 3 So log_3(9 × 27) = 2 + 3 = 5
Common mistakes to avoid
- Adding the numbers inside the logarithm instead of adding the logarithms
- Using the rule when one factor is zero or negative
- Forgetting that the same base must be used for both logarithms
Did You Know?
Logarithms turned multiplication into addition, and were called "a device that doubled the lifespan of astronomers" by cutting their calculation time.
Share this formula
More in Algebra
View allQuadratic Formula
IntermediateFinds the roots of any quadratic equation ax² + bx + c = 0.
Discriminant
IntermediateDetermines the nature of the roots of a quadratic equation.
Binomial Theorem
IntermediateExpands any power of a binomial into a sum of terms.
Logarithm Quotient Rule
IntermediateThe logarithm of a quotient equals the difference of logarithms.
Logarithm Power Rule
IntermediateThe logarithm of a power equals the exponent times the logarithm.
Change of Base Formula
IntermediateConverts a logarithm from one base to another.