Logarithms Cheat Sheet

Logarithms Cheat Sheet

Quick Revision

A logarithm answers the question "to what power must the base be raised?" By definition:

log_b x = y   means   bʸ = x        (b > 0, b ≠ 1, x > 0)

1. The Three Laws

log_b (x × y) = log_b x + log_b y
log_b (x / y) = log_b x − log_b y
log_b (xⁿ)    = n × log_b x

A product becomes a sum, a quotient becomes a difference, and a power comes down as a multiplier.

2. Standard Values

log_b 1 = 0           (b⁰ = 1)
log_b b = 1           (b¹ = b)
log_b (bⁿ) = n

3. Change of Base

log_b x = log_c x / log_c b

A useful special case: log_b x = 1 / log_x b.

4. Common and Natural Logs

• Common log log x has base 10 (used in most exam problems).
• Natural log ln x has base e ≈ 2.718.

5. Worked Example

log₂ 8      = 3          because 2³ = 8
log₁₀ 1000  = 3          because 10³ = 1000
log₃ 9 + log₃ 3 = 2 + 1 = 3

6. Exam Quick Tips

• You can only add or subtract logs that have the same base.
• log of a number between 0 and 1 is negative; log of 0 is undefined.
• Turn a power into a product with the third law before you compute.

Keep revising with the rest of the Maths section:

• All Maths topics
• Surds and Indices Cheat Sheet
• Algebra Cheat Sheet

How this was made: Drafted with AI assistance; spot-checked by the editor on 31 October 2026.