Finding inverse function of $f(x)=10^x+5^x+1$

Question

As the title of the question says , how to find $f^{-1}$ for this example ?

Of course $f$ is one-to one and with a simple transform it would be $y-1=5^x(2^x+1)$

$\Rightarrow \log_5(\dfrac{y-1}{2^x+1})=x$

Any hints for how to go further ?

Answer 1

I am afraid you can't invert this analytically.

For large positive $x$, the first term dominates and $x=f^{-1}(y)\approx\log_{10}(y)$.

For large negative $x$, the first term is neglectible and $x=f^{-1}(y)\approx\log_5(y-1)$.

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

$$y=f(x)\approx3+x(\ln(10)+\ln(5))+\frac{x^2}2(\ln^2(10)+\ln^2(5)),$$

and solve for $x$.