LogkC

LogkC, typically written as log_k(C), denotes the logarithm of C with base k. By definition, log_k(C) equals the number x such that k^x = C. The base k must be a positive number different from 1, and the argument C must be positive.

Change of base is a standard tool: log_k(C) can be computed using any common or natural logarithm. Specifically, log_k(C) = ln(C) / ln(k) = log_b(C) / log_b(k) for any base b > 0, b ≠ 1. This allows calculations in bases other than k to be performed with a calculator that provides a standard log or natural log function.

Key properties include: log_k(C^a) = a log_k(C); log_k(1) = 0; log_k(k) = 1; log_k(C1·C2) = log_k(C1) + log_k(C2); and log_k(C1/C2) = log_k(C1) − log_k(C2). The sign and magnitude of log_k(C) depend on the base k and the argument C: for k > 1, log_k(C) > 0 when C > 1 and < 0 when 0 < C < 1; if 0 < k < 1, the function is decreasing and the sign behavior reverses accordingly.

In practice, log_k(C) is used to linearize multiplicative relationships, solve exponential equations, and simplify expressions in algebra, calculus, and data analysis. It is a standard notation in mathematics, with log_k C sometimes written as log base k of C to emphasize the base.