Logarithms

log_a N = x simply means a^x = N. A logarithm is the power you must raise the base to. Unless a base is written, it is 10.

Its whole usefulness is that it downgrades every operation by one step: multiplication becomes addition, division becomes subtraction, and a power becomes a multiplication.

• log(mn) = log m + log n
• log(m/n) = log m − log n
• log(m^p) = p × log m
• log_b a = log a / log b (change of base)
• log_a a = 1 and log_a 1 = 0

Three question types:

1. Evaluate using given values of log 2, log 3, log 5 — break the number into 2s, 3s and 5s first. log 45 = log(9 × 5) = 2 log 3 + log 5.
2. Change of base, usually log_(p^m) p^n = n/m.
3. Number of digits in N = [log N] + 1, where [ ] is the whole-number part.

The single most common mistake is inventing log(m + n) = log m + log n. There is no rule for the log of a sum. The addition rule applies only to a product. Before you split anything, check that the numbers are being multiplied.

log(mn) = log m + log n · log(m/n) = log m − log n · log(m^p) = p log m · log_b a = log a/log b · digits = [log N] + 1

Given log 2 = 0.3010, find log 32. → 32 = 2^5, so log 32 = 5 × log 2 = 5 × 0.3010 = 1.5050. Sanity check: 32 lies between 10 and 100, so its log must lie between 1 and 2 — and 1.5050 does.