I don't know if this will help or not. Google didn't turn up anything useful on "exponential transformations".
There are 4 basic things you can do to transform a function.
translate it's input: $f(x-h)$
scale it's input: $f(d x)$
scale it's output: $a f(x)$
translate it's output: $k + f(x)$
Doing all 4 of these results in
$f(x) \rightarrow k+a f(d(x-h))$
It looks like your original $f(x)=b^x$ and so applying above we get