The idea
Logarithm laws turn multiplication into addition and powers into multiplication. They are useful for solving exponential equations when both sides cannot be written as powers of the same simple base. The change-of-base formula is logₐx = ln x/ln a. Keep domain restrictions in view when combining logarithms or solving equations.
Worked example
Solve 3ˣ = 20, giving the answer to three decimal places.
- Take natural logarithms: ln(3ˣ) = ln 20.
- Use the power law: x ln 3 = ln 20.
- x = ln 20/ln 3 ≈ 2.727.
A mistake to avoid
There is no law that makes log(x + y) equal to log x + log y. The sum of logarithms represents a product.
Check your understanding
Simplify log₂8 + log₂4.
Show the explanation
It is log₂(8 × 4) = log₂32 = 5. Evaluating separately also gives 3 + 2 = 5.
Explore the lesson
This guide connects to Rational exponents, log laws, and exponential equations in the IB Mathematics: Analysis and Approaches course. The first two published lessons per subject are free previews; further interactive lessons require an account.