How can one find the inverse of the funtion $f:[0, \infty) \rightarrow[0, \infty)$ defined as $f(x)=x^2\log(1+x^2)$ using $log_2$?
The general process of finding the inverse for functions that involve the e.g. $log(1+x^2)$ is clear to me. I struggle with with the $x^2\log(\cdots)$ term. I tried reformulating using logarithm rules. However, it didn't bring me further.
Any hint is highly appreciated.
Best Answer
Using $\log_2$ or using $\log$ is the same thing, and in any case you cannot invert $x^2\log(1+x^2)$ over $\mathbb{R}^+$ using elementary functions. You can by using something similar to the Lambert $W$ function, i.e. the inverse function of $xe^x$, which has a nice power series centered at the origin, by the Lagrange inversion theorem. In order to invert $x^2\log(1+x^2)$ is it enough to invert $x\log(1+x)$, or $\sqrt{x\log(1+x)}$, which meets the criteria for the application of the previous theorem. Anyway the coefficients of the power series giving a local inverse are not nice, since they depend on the derivatives at the origin of $\frac{1}{\log(1+x)^k}$, i.e. on generalized Gregory coefficients.