lnFGx

lnFGx is a notation used in mathematics to denote the natural logarithm of a quantity labeled FGx. The exact interpretation of FGx depends on context and is commonly one of two forms: the product F(x) G(x) of two functions, or the composition F(G(x)) of two functions. In all cases, the logarithm is with base e and is defined only when the inner quantity is positive.

Product interpretation: If FGx = F(x) G(x) with F(x) > 0 and G(x) > 0, then ln FGx = ln F(x) + ln G(x). This follows the standard logarithm rule ln(ab) = ln a + ln b. For example, if F(x) = x+2 and G(x) = 3x+1, then for x > -1/3 the quantity FGx is positive and ln FGx = ln(x+2) + ln(3x+1). At x = 0 this gives ln FG0 = ln 2 ≈ 0.693.

Composite interpretation: If FGx = F(G(x)), then the argument must be positive: F(G(x)) > 0. In this case ln FGx = ln(F(G(x))). There is no general simplification unless explicit forms for F and G are known. For instance, if F(t) = t^2 + 1 and G(x) = x, then FGx = x^2 + 1 > 0 for all x, so ln FGx is defined everywhere and equals ln(x^2 + 1).

Domain considerations: The central constraint is positivity of the inner quantity FGx. If FGx can be nonpositive for some x, ln FGx is undefined at those x, limiting the domain accordingly.

See also: natural logarithm, logarithm rules, function composition, product of functions.