logbc

logbC denotes the logarithm of C with base b, defined as the unique exponent y that satisfies b^y = C. It is written as log_b(C) or log_b C. The base b must be a positive real number not equal to 1, and C must be positive; otherwise the logarithm is not defined in the real numbers.

Key values and rules: log_b(1) = 0 and log_b(b) = 1. The logarithm converts multiplication into addition: log_b(xy) = log_b x + log_b y. It converts exponentiation into multiplication: log_b(x^k) = k log_b x. For division, log_b(x/y) = log_b x − log_b y. The function log_b is increasing if b > 1 and decreasing if 0 < b < 1.

Change of base: log_b C = ln C / ln b = log_k C / log_k b for any positive base k ≠ 1. This allows computing log_b C using common logs (base 10) or natural logs (base e).

Examples: log_2(8) = 3, since 2^3 = 8. log_10(1000) = 3. log_{1/2}(4) = -2, since (1/2)^(−2) = 4.

Common bases and notation: base 10 is often written as log, base e as ln, and base b explicitly as log_b. In some contexts, log b C or log b(C) are used, but log_b(C) is the standard mathematical notation.

Applications include data that span orders of magnitude, exponential growth and decay modeling, and various scientific and engineering computations. Negative numbers and zero are outside the real domain of logarithms.