<s>[INST] Inverse of a non-trivial exponential function I am asked to determine the inverse function of this function,
    $$f(x)=2^{x}+3^{x}$$
The inverse function can not be found explicitly, since there is no way to explicitly clear x, but this does not mean that it has no inverse.
I could show a way to find the inverse of this function
Edit  How can we find the inverse of this function with the problems that it presents in the clearance of the x?
 [/INST] You could, for example, write the solution to $2^x + 3^x = y$ as a series in powers of $y-2$:
$$\eqalign{x &= \frac{y-2}{\ln(6)} - \left(\ln(3)^2 + \ln(2)^2\right) \frac{(y-2)^2}{2 \ln(6)^3}\cr +& \left(2 \ln(3)^4 - \ln(3)^3\ln(2)+6\ln(3)^2\ln(2)^2-\ln(3)\ln(2)^3 + 2 \ln(2)^4\right)\frac{(y-2)^3}{6 \ln(6)^5}\cr - &\left(3 \ln(3)^6 - 4 \ln(3)^5 \ln(2) + 18 \ln(3)^4 \ln(2)^2 - 10 \ln(3)^3 \ln(2)^3 + 18 \ln(3)^2 \ln(2)^4 - 4 \ln(3)\ln(2)^5 + 3 \ln(2)^6 \right) \frac{(y-2)^4}{12 \ln(6)^7}
\cr + &\ldots }$$
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