logbaseb

Log base b, denoted log_b x, is the logarithm with base b. It is the inverse function of the exponential function b^y. For positive x and base b > 0 with b ≠ 1, the equalities b^{log_b x} = x and log_b(b^y) = y hold.

Domain and monotonicity: x must be positive; the base b must satisfy b > 0 and b ≠ 1. When b > 1, log_b x is increasing; when 0 < b < 1, it is decreasing.

Definition and change of base: log_b x can be defined as ln x / ln b, where ln denotes the natural logarithm. The change-of-base formula states log_b x = log_k x / log_k b for any positive base k ≠ 1 (commonly k = e or k = 10).

Properties: log_b(1) = 0, log_b(b) = 1. log_b(xy) = log_b x + log_b y for x, y > 0. log_b(x^r) = r log_b x for any real r.

Examples: log_2 8 = 3; log_10 100 = 2. Using natural logs: log_2 8 = ln 8 / ln 2 ≈ 3.000.

Relation to other logarithms and uses: ln x is log_e x; log_10 x is log base 10 of x. Log bases are used in solving exponential equations, in growth and decay models, and in computer science, particularly for measuring quantities in binary scales. The concept generalizes to any positive base b ≠ 1.