LogarithmsBoth directions
Name: Date:
Work out each logarithm, or find the missing number.
1log₅ 125 =
2log₃ 19 =
3log₂ 256 =
4log₃ 729 =
5log₄ = 2
6log₁₀ 1000 =
7log₉ = 2
8log₂ = 8
9log₃ = 2
10log₅ = 2
Logarithms worksheets
A logarithm asks one question: which power of the base lands on this number? These free printable worksheets ask it in both directions — the logarithm of a number, and the number whose logarithm it is — and the answer key prints the power that settles it.
Kid stuck on one? You don’t just get the sheet.
Logarithms
The answer key prints on a separate page.
What's on this sheet
The notation is the whole difficulty. `log₂ 32` is not an instruction to divide by 2; it is a question about exponents, and a student who reads it as an operation gets 16. Every walk on these sheets works forwards from the power — 2⁵ = 32, therefore log₂ 32 = 5 — because that is both how the questions were built and how a person answers one.
Why this sheet works
The reciprocal questions are where the standard actually lives. `log₂ 1/32 = −5` is the fact that a logarithm goes negative exactly when the number is between 0 and 1, and a student who has only ever seen whole arguments reads a negative logarithm as their own mistake. Roughly a quarter of every mixed sheet is a unit fraction for that reason, and the fraction is printed stacked rather than with a slash so it cannot be misread as a division.
How this one works
One question from this sheet, worked through a step at a time.
log₅ 125
5³ = 125
log₅ 125 = 3
- log₅ 125 asks a question: which power of 5 lands on 125? It is not an instruction to divide by 5 — that is the mistake this notation collects.
- So log₅ 125 is 3.