logarithmic

Logarithmic refers to logarithms, the inverse functions of exponentiation. For a base b > 0 with b ≠ 1, the logarithm of a positive number x is written log_b(x) and is defined by log_b(x) = y if and only if b^y = x. The natural logarithm uses base e and is denoted log_e(x) or ln(x).

Key identities follow from the laws of exponents. log_b(xy) = log_b x + log_b y, and log_b(x^k) = k log_b x. The change of base formula is log_b x = log_k x / log_k b for any positive base k ≠ 1. These properties allow solving exponential equations and transforming multiplicative relationships into additive ones.

Calculus interacts with logarithms as well. The derivative of log_b x is 1/(x ln b). The integral ∫ log_b x dx = x log_b x − x/ln b + C. The natural log ln x has the series expansion ln(1+x) = x − x^2/2 + x^3/3 − … for -1 < x ≤ 1 (x ≠ -1).

Behavior and graph. For base b > 1, log_b x is increasing; for 0 < b < 1, it is decreasing. The graph is defined on x > 0, with a vertical asymptote at x = 0 and a point (1, 0). The function crosses the point (e, 1) when b = e.

Applications and scales. Logarithms compress large ranges of values and underpin logarithmic scales used in science and engineering, such as pH, decibels, and the Richter scale. They also linearize exponential growth, aiding data analysis, modeling, and solving equations involving exponential functions.