How to calculate the inverse function of $y = x + \ln(x)$?

Question

I have had incidents in the past where my teacher gives a question on non-calculator practice exams that are impossible to solve without a calculator, where you are stuck in an endless loop of moving logs and e's around with out going anywhere.

Without a calculator I am stuck at something like $e^{x-y} = y$

However, I can't even seem to solve this with a regular calculator, and was wondering, is it even possible to solve with a calculator?

Ty.

Answer 1

As far, as I know it is not expressible in elementary functions.

Look up, for example Lambert's W-function.

You can use this.

Lambert's function calculator

Answer 2

$$e^y=e^{x+\ln x}$$

$$e^y=e^x\cdot e^{\ln x}=xe^x$$

The inverse of $xe^x$ is Lambert's W-function

Then

$$x=W(e^y)$$

Answer 3

As it was already pointed out $$ y = x + \ln(x) \iff e^y = x e^x $$ which is the inverse of the Lambert W function, which is known to be non-elementary.

So you would need to go through some approximation with your calculator.

E.g. if you got some value $y$ you could try Newton-Raphson to find a root of the equation $$ f(x) = x e^x - e^y $$ which turns into the iteration \begin{align} x_{n+1} &= x_n - \frac{f(x_n)}{f'(x_n)} \\ &= x_n - \frac{x_n e^{x_n} - e^y}{e^{x_n} + x_n e^{x_n}} \\ &= x_n - \frac{x_n e^{x_n} - e^y}{x_n e^{x_n} + e^{x_n}} \\ \end{align}