# Does the function $\log(1+\exp(x))$ have a conventional name?

Does the function $\log(1+\exp(x))$ (or the function $\log(1+\exp(-x))$) have a conventional or at least fairly common name?

Alternatively, is it closely related to some reasonably well-known, named function?

[It arises in a problem I'm working on and it would be nice to use a reasonably standard terminology and notation rather than either writing $\log(1+\exp(g(x)))$ each time (where $g$ is a fairly complicated set of terms), or coming up with a non-standard notation for it (like $\omega(g(x))$ say) and then discovering I could have used a symbol and term for it that are already recognized). It would also help in terms of locating properties, should I end up needing any beyond the simple ones I've already derived.]

• May I propose $\text{Glen_b}_\text{+}$ and $\text{Glen_b}_\text{-}$ ? Back to serious, I do not think that they have a name. Mar 13, 2016 at 8:43
• “As we shall be often using the following function, we shall write $\operatorname{wow}x=\log(1+\exp x)$” Mar 13, 2016 at 10:00
• @egreg yes, thanks, I had already written something rather like that, but I'd rather not dodge convention if I can find one. Mar 13, 2016 at 10:11
• @norbert No, it isn't, the sigmoid function is bounded above while this function is not. It is related to it - it's minus the log of the sigmoid function. Apr 3, 2016 at 23:39
• Last year, in a GLM course I took, we called $x \mapsto \log(1 + e^x)$ the "soft positive part" function since it's roughly $\max\{0,x\}$. A quick google search didn't reveal any references calling it, though. Sep 3, 2017 at 19:06