TITLE: How to invert this exponential function to solve for x:  $y = a \exp(bx) + c \exp(dx)$?
QUESTION [5 upvotes]: Cheers.
So if I don't make sense, I have a value for $y$, I need to know what $x$ is.
$$y = a \exp(bx) + c \exp(dx)$$
$a =       12.85$,
$b =    0.001857$,
$c =      -54.24$,
$d =    -0.05316$

REPLY [2 votes]: You have $y=ae^{bx} + ce^{dx}$ with $a, b$ positive and $c, d$ negative. So this means:

if $x$ is large and positive, then $y \approx ae^{bx}$
if $x$ is large and negative, then $y \approx ce^{dx}$

So for $x$ large and positive, $x \approx (1/b) \log (y/a)$; for $x$ large and negative, $x \approx (1/d) \log (y/c)$ -- these come from solving the above approximations for $x$. If you need to go further I'd say start with these approximations and then use something like Newton's method.