lnxy

lnxy commonly denotes the natural logarithm of the product of x and y, written as ln(xy). The natural logarithm is the logarithm with base e, where e is approximately 2.71828. The function is defined for real numbers when the product xy is positive, i.e., xy > 0. The standard identity is ln(xy) = ln x + ln y, which holds for x > 0 and y > 0. If either x or y is nonpositive, the real logarithm of xy is undefined; complex extensions exist but go beyond the real-valued setting.

This logarithmic rule enables the simplification of products inside logarithms and is a cornerstone of many algebraic and calculus techniques. Related properties include ln(x^k) = k ln x for x > 0, and the fact that the natural logarithm is the inverse function of the exponential e^x, so e^{ln t} = t and ln(e^t) = t for t > 0. The derivative of the natural logarithm is d/dx [ln x] = 1/x for x > 0, and its antiderivative is ∫(1/x) dx = ln|x| + C, highlighting its behavior near zero and across sign changes.

Examples help illustrate the rule: with x = 2 and y = 8, ln(xy) = ln(16) ≈ 2.772589, and ln x + ln y ≈ 0.693147 + 2.079442 = 2.772589. The identity underpins many practical calculations in math, science, and engineering whenever products are analyzed in logarithmic form.