Approximating Lambert W-function as a fraction of logarithm I need to solve an equation and I was able to find a solution in terms of Lambert W-function. However I need to solve the same equation for another set of parameters. I need to compare the root of those two equations but the relation is too complicated, The major complexity is posed because of Lambert W-function. Therefore I need to find a way to approximate $W(x)$ with logarithm.
I know the results that $W(x) \approx \ln x - \ln \ln x + \frac{\ln \ln x}{\ln x}+\cdots$ and also I know that
$$\lim_{x\to \infty} \frac{W(x)}{\ln x} = 1$$
However non of these are helpful.
I was wondering is there anyway I can find a relation like $W(x) \approx \alpha \ln x$ for a known $\alpha$ when $x$ is large but bounded?
I am interested in $x \in (10^3,10^4)$
 A: In order to get the approximation you look for, consider the norm
$$\Phi(\alpha)=\int_{10^3}^{10^4} (W(x)-\alpha  \log (x))^2\, dx$$
The definite integral is known. Now, differentiating $\Phi(\alpha)$ with respect to $\alpha$ we have a nasty linear equation; numerically, its solution is $\alpha=0.778164$.
The simplest would be a linear regression. Done, it leads to the same parameter and $R^2=0.999941$
$$\begin{array}{clclclclc}
 \text{} & \text{Estimate} & \text{Standard Error} & \text{Confidence Interval} \\
 \alpha & 0.778164 & 0.000063 & \{0.778040,0.778287\} \\
\end{array}$$
Edit
We can improve the approximation adding a linear term. So, now the norm
$$\Psi(\alpha,\beta)=\int_{10^3}^{10^4} (W(x)-\alpha  \log (x)-\beta)^2\, dx$$
All required antiderivatives are known. Differentiating with respect to $\alpha$ and $\beta$ leads to two linear equations from which
$$\alpha=0.864126\qquad \text{and} \qquad \beta=-0.731281$$
The improvement is interesting
$$\Phi_{\text{min}}=23.1588\quad \text{and} \quad \Psi_{\text{min}}=0.0740 \quad\implies\quad \frac{\Phi_{\text{min}} } {\Psi_{\text{min}} }=313$$
