What a logarithm means
If aˣ = y then loga y = x. The logarithm answers the question: to what power must the base a be raised to give y? Here a > 0 and a ≠ 1, and y > 0.
Key idea
Product: loga(xy) = loga x + loga y. Quotient: loga(x/y) = loga x − loga y. Power: loga(xⁿ) = n loga x. Also loga a = 1 and loga 1 = 0.
Worked example
Evaluate log₃ 45 − log₃ 5. Using the quotient law this is log₃(45 ÷ 5) = log₃ 9. Since 3² = 9, log₃ 9 = 2.
Change of base
To evaluate a logarithm with your calculator or to combine different bases, use loga b = (log b) / (log a), where the right-hand logs are to any convenient base (often base 10).
Remember
- You can only add or subtract logs with the same base.
- lg means log to base 10.
- Bring a coefficient inside as a power: n log x = log(xⁿ).