THE PRODUCT RULE FOR
LOGARITHMS
WHAT IS THE PRODUCT RULE FOR
LOGARITHMS?
Simply stated, the product rule for logarithms is
this:
logb(xy) = logb(x) + logb(y)
 This rule can prove very useful for simplifying
logarithms.

PROOF OF THE PRODUCT RULE
Let M = bx and let N = by.
 Then logb(M) = x and logb(N) = y.
 In addition, MN = bx+y by the properties of
exponents.
 Taking the base b logarithm of both sides of our
previous equation, we get logb(MN) = x + y.
 Substituting in logb(M) for x and logb(N) for y, we
get logb(MN) = logb(M) + logb(N).

EXAMPLE
We can use the product rule to solve many
logarithm problems.
 For example, what is log10(125) + log10(8)?
 Using the product rule,
log10(125) + log10(8) = log10(125 * 8)
 log10(125 * 8) = log10(1000) = 3.

EXAMPLE
What is log3(6) – log3(2)?
 Using the product rule, log3(6) = log3(3) + log3(2).
 Thus, we have log3(3) + log3(2) – log3(2), which is
equal to log3(3), which, of course, is 1.
