Asymptotic expansion to solution of $x - \log x = a$ for large $a$

Question

Suppose $a \in \mathbb{R}$ is very large. Then there are two solutions to $x - \log x = a$. I was wondering what the asymptotic expansion of the larger solution to this equation is. The first term must be $a$ but what are the lower order terms?

Answer 1

Let $x=a+b$, where $b\ll a$. Then

$$a=a+b-\log(a+b)=a+b-\log a-\log\left(1+\dfrac ba\right)\approx a+b-\log a-\dfrac ba$$

and

$$x\approx a+\frac{a\log a}{a-1}.$$

In blue, the exact curve.

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You can also address the problem with good ol' Newton's iterations,

$$x\leftarrow x-\frac{x-\log x-a}{1-\dfrac1x}=x+\frac{x(\log x+a-x)}{x-1},$$

giving the approximations

$$a,\\ a+\frac{a\log a}{a-1},\\ a+\frac{a\log a}{a-1}+\frac{\left(a+\dfrac{a\log a}{a-1}\right)\log a}{a+\dfrac{a\log a}{a-1}-1},\\ \cdots$$