A problem I have gives a function of the form $$f(x) = \frac{C_1a_1^x+C_2a_2^x}{C_3a_3^x+C_4a_4^x}$$ where $a_i$ and $C_i$ are real constants. When I play around with it and plot it for various random values of $a_i$ and $C_i$ and most of the time it fits the exponential function of the form $ce^{dx}$ spot on (i.e. $R^2=1$ in a least squares regression).
Is there a way to express $f(x)$ as $ce^{dx}$ where $c$ and $d$ depends on $a_i$ and $C_i$?
I have tried the tricks I know but none of them seem to work.
To express $f(x)$ as $ce^{dx}$, usually no. To approximate it, often yes. For example, for large $x$, look at the four exponentials, and the bigger ones win.
For example, $$ \frac{5\;2^x - 6\;4^x}{2^x + 3\;3^x} \sim -2\;\left(\frac43\right)^x\quad \text{as } x \to \infty $$ which we do by noting $4^x$ defeats $2^x$ in the numerator, and $3^x$ defeats $2^x$ in the denominator: $$ \frac{5\;2^x - 6\;4^x}{2^x + 3\;3^x} = \frac{5\;(2/3)^x - 6\;(4/3)^x}{(2/3)^x + 3\;(3/3)^x} \sim \frac{0-6\;(4/3)^x}{0+3} = -2\;\left(\frac43\right)^x $$