The idea
A logarithm answers “what power of the base gives this number?” For example, log₂8 = 3 because 2³ = 8. Common logarithms use base 10; natural logarithms use base e and are written ln. Negative logarithm values are allowed: log₁₀0.01 = −2 because 10⁻² = 0.01.
Worked example
Solve 2ˣ = 32, then evaluate log₄(1/16).
- 32 = 2⁵, so x = 5.
- Since 1/16 = 4⁻², log₄(1/16) = −2.
- Check each answer by rewriting it in exponential form.
A mistake to avoid
A negative result is allowed, but a logarithm of a negative argument is not defined in the real-number setting.
Check your understanding
Find log₅1 and explain your answer.
Show the explanation
log₅1 = 0 because 5⁰ = 1.
Explore the lesson
This guide connects to Integer exponents and intro to logarithms in the IB Mathematics: Analysis and Approaches course. The first two published lessons per subject are free previews; further interactive lessons require an account.