Logarithmic functions

The definition of a logarithm is a rewritten exponential: y = log_b(x) means exactly the same thing as b^y = x, for a base b > 0 with b ≠ 1. So log₂(8) = 3 because 2³ = 8, and log₁₀(0.01) = −2 because 10⁻² = 0.01. Two bases are common enough to get their own notation: log x with no subscript means base 10 (the common log), and ln x means base e ≈ 2.718 (the natural log).

Because it inverts an exponential, the graph of y = log_b(x) is the reflection of y = b^x across the line y = x. Its domain is x > 0 — you cannot take the log of zero or a negative number in the real numbers — while its range is all real numbers. Every logarithmic graph passes through (1, 0), since b⁰ = 1, and has a vertical asymptote at x = 0. For b > 1 the function increases, but slowly: it takes a tenfold increase in x to move log₁₀(x) up by 1, which is why logarithmic scales compress quantities that span many orders of magnitude, such as pH, decibels, and earthquake magnitude.

Three properties convert products and powers into sums and multiples, which is what makes logs useful for solving equations. logb(mn) = logb(m) + logb(n), logb(m/n) = logb(m) − logb(n), and logb(m^p) = p · logb(m). The change-of-base formula, log_b(x) = ln(x) / ln(b), lets you evaluate any base on a calculator. To solve an equation such as 3^x = 20, take the log of both sides and apply the power rule: x = ln(20) / ln(3).

Logarithms appear across several exams. The AMC 12 tests logarithmic manipulation in its advanced algebra problems, often requiring clever use of the product and power rules. AP Calculus AB covers the special limits and derivatives tied to logs — the derivative of ln x is 1/x, and logarithmic differentiation simplifies messy products and powers. The FE Mechanical exam includes logarithms in its analytic geometry and mathematics section, where the reference handbook supplies the identities.