lnGx

lnGx, read as the natural logarithm of G evaluated at x, is commonly written ln(G(x)). It denotes the logarithm of a real-valued function G at the input x. For the real-valued logarithm to be defined, G(x) must take positive values: G(x) > 0. In complex analysis, ln(G(x)) can be defined for nonzero complex values but involves branch choices.

Domain and basic properties: The domain of ln(G(x)) is the set of x for which G(x) > 0. If G is differentiable and positive, ln(G(x)) is differentiable as well and obeys the chain rule: d/dx [ln(G(x))] = G′(x) / G(x). More generally, higher-order derivatives can be expressed in terms of G and its derivatives, for example the second derivative is (G″/G) − (G′/G)².

Examples: If G(x) = x + 2, then lnGx = ln(x + 2) for x > −2. If G(x) = e^x, then lnGx = ln(e^x) = x identically. For G(x) = x^2 + 1, the expression lnGx is defined for all real x, with derivative 2x/(x^2 + 1).

Extensions and notes: The concept can extend beyond scalar G. If G(x) is a positive definite matrix-valued function, one may consider the matrix logarithm logm(G(x)) in contexts such as differential geometry or multivariate statistics. In such cases, care is needed regarding domains and branches. In general usage, lnGx remains a shorthand for ln(G(x)) with the usual interpretation and caveats.