Introduction to Logarithms

A logarithm answers: what exponent turns the base into this number?

log₂(8) = 3, because 2³ = 8.
log₁₀(1000) = 3, because 10³ = 1000.

The definition to hold onto:
log_b(a) = c ⟺ bᶜ = a

Converting between those two forms handles most problems.

Two conventions: log with no base written means base 10, and ln means base e (about 2.718).

The domain matters. You can only take the log of a positive number, because bᶜ is always positive. log(0) and log(−5) do not exist.

The logarithm log_b x is the inverse of the exponential b^x for b > 0, b ≠ 1. As inverses, log_b(b^x) = x and b^{log_b x} = x, and their graphs reflect across y = x.

Consequently log_b has domain (0, ∞) and range ℝ, with a vertical asymptote at x = 0 — the mirror of the exponential's horizontal asymptote at y = 0. This is why the argument of a logarithm must be positive, and why solving log equations requires domain checks.

Base conventions: log x is base 10 in most contexts, ln x is base e, and lg or log means base 2 in computer science.

The change of base formula log_b x = (ln x)/(ln b) allows any base to be evaluated with a calculator and shows that all logarithmic functions are constant multiples of each other.

Logarithms compress multiplicative scales into additive ones, which is why decibels, pH, the Richter scale and stellar magnitudes are all logarithmic: they turn ratios spanning many orders of magnitude into a readable range.