logexpx

Logexpx is a mathematical expression referring to the composition of the natural logarithm with the exponential function, commonly written as log(exp(x)) or logexpx. In real analysis, it simplifies to the identity function on the real numbers, because exp(x) > 0 for all real x and ln(exp(x)) = x. Consequently, for every real x, logexpx(x) = x. The function is thus monotonically increasing and differentiable everywhere on the real line, with derivative d/dx log(expx) = exp(x)/exp(x) = 1, and second derivative 0.

From a functional perspective, logexpx is its own inverse on the real line, since log(expx) = x implies that its inverse is also the identity. By contrast, the inner and outer components are inverse to each other in the two-function sense only when considered as the pair log and exp individually; the pair are inverses of each other, not log∘exp.

Complex domain caveat: for complex arguments, exp(z) is periodic in the imaginary direction and the complex logarithm is multi-valued; log(exp(z)) equals z modulo 2πi, depending on branch choices. In practical numerical contexts, logexpx is used mainly as a reminder that the log of an exponential can cancel to the original argument for real inputs, and it has limited use as a distinct function.

Related functions and variants include log1pexp, which denotes log(1 + exp(x)), and is used in convex analysis and machine learning, as well as the log-sum-exp trick used for numerical stability in log-likelihood computations.