Product rule of logarithms worksheets

Adding two logarithms multiplies their numbers. These free printable worksheets take that one law on its own, and every question asks for a sum to be written as a single logarithm.

Product rule of logarithms
Product rule of logarithms worksheet — Bases 2–10, free printable with answer key

The answer key prints on a separate page.

What's on this sheet

The law is a fact about exponents wearing other clothes. Multiplying two powers of the same base adds their exponents, and a logarithm is an exponent, so adding logarithms has to multiply the numbers underneath. Saying that once is worth more than memorizing the line, because it also explains why the bases have to match.

Why this sheet works

The mixed sheet is where a student learns to choose between the three laws. This one comes first, for a student who still needs to see the law work a few times in a row. Each answer is a single logarithm, so the marking is a quick glance and any slip stands out.

How this one works

log₅ 2 + log₅ 8

log₅ 2 + log₅ 8
log₅ m + log₅ n = log₅ (m × n)
2 × 8 = 16
log₅ 2 + log₅ 8=log₅ 16
  1. Both logarithms are to the same base, 5. Only logs with the same base can be combined.
  2. Adding two logs with the same base gives the log of the two numbers multiplied.
  3. So the two arguments become one: 2 × 8 = 16. The answer is log₅ 16.