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 $x \in (10^3,10^4)$

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 $$\Phi_{\text{min}}=23.1588\quad \text{and} \quad \Psi_{\text{min}}=0.0740 \quad\implies\quad \frac{\Phi_{\text{min}} } {\Psi_{\text{min}} }=313$$