Logarithms, step by step
A logarithm asks one question: which power of the base lands on this number?
The question
What is log₉ 1/81?
log₉ 1/81
9² = 81
9⁻² = 1/81
log₉ 1/81 = −2
- log₉ 1/81 asks a question: which power of 9 lands on 1/81? It is not an instruction to divide by 9 — that is the mistake this notation collects.
- 1/81 is 1 over 81, and a negative exponent is what writes a reciprocal: 9⁻² means 1 ÷ 9². So the power that lands on 1/81 is the same one, turned over.
- So log₉ 1/81 is −2. A logarithm is negative exactly when the number is between 0 and 1 — nothing has gone wrong.
How it works.
- 01
log₉ 1/81 asks a question: which power of 9 lands on 1/81? It is not an instruction to divide by 9 — that is the mistake this notation collects.
- 02
1/81 is 1 over 81, and a negative exponent is what writes a reciprocal: 9⁻² means 1 ÷ 9². So the power that lands on 1/81 is the same one, turned over.
- 03
So log₉ 1/81 is −2. A logarithm is negative exactly when the number is between 0 and 1 — nothing has gone wrong.