lnFG

lnFG is a compact notation encountered in mathematics and related fields. It most often refers to the natural logarithm of the product FG, where F and G denote positive real numbers or positive-valued functions and FG represents their product.

Under this interpretation, ln(FG) = ln F + ln G, provided F > 0 and G > 0. If F and G are functions of a common variable x, then ln(FG(x)) = ln(F(x)G(x)) = ln F(x) + ln G(x) wherever F(x) > 0 and G(x) > 0. This follows from the general logarithm rule ln(ab) = ln a + ln b for positive a and b.

Notation note: lnFG without parentheses can be ambiguous, as it could be read as ln(F)G or as ln(FG). Mathematicians typically write ln(FG) or log(FG) to avoid ambiguity, and in more formal contexts one may also specify the arguments explicitly as ln(FG) or ln(F(x)G(x)).

In calculus and analysis, the derivative of ln(FG) with respect to a variable x, assuming F and G are differentiable and positive, is (d/dx) ln(FG) = (F'G + FG')/(FG) = F'/F + G'/G. This uses the product rule and the chain rule, together with the basic logarithm identity.

Historically, the natural logarithm is denoted by ln, a notation popularized in the 18th century, following earlier development of logarithms by Napier and subsequent formalization by Euler and others.

See also: logarithm, natural logarithm, logarithmic properties.