How do I solve the following, analytically?
$$x=\log^e{(x+1)}$$
It looks like it should be simple, but whether I take the $e$th root of each side or take the $\log$ of each side (ending up with a $\log\log$), I get stuck—and both approaches seem naïve, anyhow.
It appears to have two solutions:
In case it raises some eyebrows, I only stumbled across the formula while blindly tinkering with reward functions for a productivity gamification system. In short, I needed a function $g(x)$ that behaves very similarly to $f(x)=x$ below a certain constant, e.g. 27 (which I can scale arbitrarily), then tapers off and flatlines. $g(x)=\log^e{(x+1)}$ happened to work perfectly, but of course I wouldn't assert the formula to be meaningful in any natural sense.
Try the Lambert function, this is for such relations: http://en.wikipedia.org/wiki/Lambert_W_function