Inverse of $f(x)=3^x+2^x$

Question

I'm tring to find inverse of $f(x)=3^x+2^x$ but I don't have any clue. I tried to $$y=2^x((3/2)^x+1)$$ $$\ln y=\ln2^x+\ln((3/2)^x+1)$$ $$\ln y= x \ln2+\ln((3/2)^x+1)$$ but I can't continue

Answer 1

There is no simple closed form for the inverse.

Looking at a plot of the logarithm, $\ln(3^x+2^x)$, you get a very good approximation of $x\ln(2)$ in the negatives and $\ln(3)$ in the positives, as one term quickly dominates the other (their ratio drops below $1\%$ for $|x|>12$).

For small $x$, you can use the Taylor development to a limited degree

$$3^x+2^x\approx2+(\ln(3)+\ln(2))x+(\ln^2(3)+\ln^2(2))\frac{x^2}2+(\ln^3(3)+\ln^3(2))\frac{x^3}{3!}+\cdots$$

and invert the polynomial (doable, but tedious, up to degree four; as it is just an approximation, one can wonder if it is worth the effort.)

Given the smoothness of the function, I guess that inverse cubic spline interpolation on $\ln(3^x+2^x)$ can do marvels.