lnFxGx

lnFxGx is a notation used in mathematics to denote the natural logarithm of the product of two functions evaluated at a common argument x. The standard interpretation is ln(F(x) G(x)), the natural log of the product of F and G at x. Because of log rules, when F(x) and G(x) are positive, this quantity equals ln(F(x)) + ln(G(x)).

Notation can vary, and without parentheses the meaning may be ambiguous. Some authors might write ln F(x) G(x) to indicate a different expression, such as ln(F(x)) times G(x). To avoid confusion, many texts adopt the explicit form ln(F(x) G(x)) or specify the intended meaning.

Domain and differentiation: If F and G are differentiable and positive on an interval, then ln(F(x) G(x)) is differentiable there, and its derivative is F′(x)/F(x) + G′(x)/G(x) by the logarithm product rule. If either F(x) or G(x) is not positive, the real-valued logarithm is undefined; in such cases one may restrict the domain or use a complex logarithm.

Examples: If F(x) = e^x and G(x) = x^2 + 1 (which is positive for all real x), then ln(F(x) G(x)) = ln(e^x (x^2 + 1)) = x + ln(x^2 + 1).

Applications: The expression commonly appears in probability, statistics (log-likelihoods for products of probabilities), information theory, and multiplicative models, where taking logs converts products into sums and simplifies analysis.

See also: logarithm, natural logarithm, product rule, logarithm properties.