What does the log-sum-exp function represent, and how do its properties relate to the laws of logarithms?

The log‑sum‑exp function takes a set of numbers, exponentiates each one, adds them up, and then takes the logarithm of that sum, so it is essentially \(\log(\sum_i e^{x_i})\). Because exponentiation and logarithm are inverse operations, the function behaves like a smoothed maximum: if one \(x_i\) is much larger than the others, the result is close to that largest value, just as \(\log(e^{x_{\max}})=x_{\max}\). Its properties mirror logarithm laws: \(\log(a)+\log(b)=\log(ab)\) becomes \(\log(\sum e^{x_i})\approx \max_i x_i + \log(1+\sum_{j\neq i} e^{x_j-x_i})\), showing how the sum inside the log distributes over the exponentials. For example, with \(x_1=2\) and \(x_2=0\), the function gives \(\log(e^2+e^0)\approx \log(7. 389+1)=2.