Ahhh. Thank you. That makes a lot of sense. I assume b is a constant, then I solve for b as a function of x! I am pretty dense.

Anyways, here's something else I found that may be of use,

Let's say that .

Then by definition

Now substitute g(x) for x,

So, in conclusion, the half-iterate of e^x is the exponential of the half-iterate of ln(x).

Not sure if that is useful but I think it's kinda neat. Hopefully I didn't make another mistake, but it's always possible.

Anyways, here's something else I found that may be of use,

Let's say that .

Then by definition

Now substitute g(x) for x,

So, in conclusion, the half-iterate of e^x is the exponential of the half-iterate of ln(x).

Not sure if that is useful but I think it's kinda neat. Hopefully I didn't make another mistake, but it's always possible.