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

$ab^{(d(x-h))}+k$