lnc

lnC denotes the natural logarithm of a quantity C, with ln representing the logarithm to base e. For a real-valued lnC to be defined without extending to complex numbers, C must be positive. In many mathematical and scientific contexts, C is treated as a positive constant or a positive variable, and lnC is used to linearize exponential relationships.

Formally, lnC = log_e(C). Basic properties follow from the laws of logarithms: ln(C1 C2) = lnC1 + lnC2; ln(C^k) = k lnC; ln(1) = 0. The function lnC is strictly increasing on (0, ∞) and its derivative with respect to C is 1/C. Limits include lim_{C→0+} lnC = -∞ and lim_{C→∞} lnC = ∞. The natural logarithm is the inverse of the exponential function: e^{lnC} = C and ln(e^x) = x.

Notation varies: ln(C) is standard, but some authors write lnC without parentheses. When C ≤ 0, a real-valued lnC does not exist; in complex analysis, lnC can be defined with a complex value, involving the magnitude and an argument (ln|C| + i Arg(C)) and is multi-valued due to the Arg.

Applications of lnC appear across disciplines, including modeling population growth and decay, continuous compounding in finance, and as a data transformation in statistics to stabilize variance or linearize multiplicative effects. Examples: ln(1) = 0; ln(10) ≈ 2.302585.