Mapping intervals exponentially

I have two intervals: $X=[A,B]$ and $X'=[C,D]$.

If I'd like to map $X$ to $X'$, I usually use this equation:

$$f(t)= \frac{D-C} {B-A} t + \frac{(BC - AD)} {B-A}$$

where $t$ is the time.

However this is linear mapping. I would like to do the mapping exponentially. Could someone help me in this?

I appreciate you help.

-
I just reformatted your equations. You might want to check them over to make sure I didn't screw them up somehow. :) – Micah Mar 28 '13 at 0:18

I assume by "exponentially" you mean you want a map of the form $f(t)=ke^{rt}$. So we get the equations $$C=ke^{Ar},\qquad D=ke^{Br}$$ Then $$C/D=e^{(A-B)r}$$ so $$r={\log C-\log D\over A-B}\tag1$$ and $k=Ce^{-kr}$ where $r$ is given by (1).
EDIT: As Ross points out in the comments, that was meant to be $k=Ce^{-Ar}$.
@user951793: I believe $k=Ce^{-Ar}$ Once you have $k,r$ you have everything you need for $f(t)$ – Ross Millikan Mar 28 '13 at 0:49
@user951793: that is because you are trying to take the log of zero with $C=0$. The way $f(t)$ is defined, it cannot be zero. – Ross Millikan Mar 28 '13 at 13:04