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Quotient rule of logarithms worksheets

Subtracting two logarithms divides their numbers, first over second. These free printable worksheets practice that one law on its own, and every question asks for a difference to be written as a single logarithm.

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Quotient rule of logarithms
Quotient rule of logarithms worksheet — Bases 2–10, exact quotients, free printable with answer key

The answer key prints on a separate page.

What's on this sheet

Order is the whole risk here, and it is the one thing the product rule never asked about. log 18 − log 6 is log 3, and the same two logarithms in the other order are not. A student who has practiced only sums will sometimes divide whichever way gives a tidier number.

Why this sheet works

Every division on these sheets comes out exact, so there is no fraction to manage and the page stays on the law. Once it feels routine, the mixed sheet puts the three laws side by side and asks which one each question needs.

How this one works

log₃ 18 − log₃ 6

log₃ 18 − log₃ 6
log₃ m − log₃ n = log₃ (m ÷ n)
18 ÷ 6 = 3
log₃ 18 − log₃ 6=log₃ 3
3 = 3¹
log₃ 3=1
  1. Both logarithms are to the same base, 3. Only logs with the same base can be combined.
  2. Subtracting two logs with the same base gives the log of the first number divided by the second.
  3. So the two arguments become one: 18 ÷ 6 = 3. The answer is log₃ 3.
  4. 3 is 3 to the power 1, so this one goes further: log₃ 3 is 1.

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