Exponentials and logarithms

A logarithm answers the question “what power?”. log₂ 8 = 3 because 2³ = 8. Logs undo exponentials, so they solve equations where the unknown is a power. The natural log, ln, uses the special number e ≈ 2.718.

Key facts

  • logₐ b = c means aᶜ = b.
  • log a + log b = log ab; log a − log b = log (a/b); log aⁿ = n log a.
  • ln is log base e; ln(eˣ) = x.
  • To solve aˣ = b, take logs: x = log b / log a.

Worked examples

Example 1

Solve 3ˣ = 50.

  1. log 3ˣ = log 50 x log 3 = log 50
  2. x = log 50 / log 3 ≈ 3.561

Answer: x ≈ 3.56

Example 2

Write 2 log 3 + log 4 as a single log.

  1. 2 log 3 = log 9
  2. log 9 + log 4 = log 36

Answer: log 36