How to solve a non-linear equation

Question

I want to solve an equation $$\log \left(\frac{b}{y}\right)=\left(\frac{x}{y}-a\right)^{\beta}.~~~~~~~~~(1)$$

Solve[Log[b/y] == (x/y - a)^\[Beta], y]

Here $b$, $x$, and $a$ are constants (In principle, we have $x/y\simeq a$). I want to find the solution $y=?$

If $a=0$, the solution is $$y^*=\left(-\frac{\beta x^{\beta}}{W\left(\beta\left(-b^{-\beta}\right) x^{\beta}\right)}\right)^{1/\beta},~~~~~~~~~(2)$$ where $W$ is the ProductLog function in MATHEMATICA.

I try to use $y=y^*+c*x^\gamma$ to find the correction term, but I failed. I guess that the correction term is not like $c*x^\gamma$.

How can I find the correction term? For my problem, $a$ is important and can be ignored.

Thanks!

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In principle, this problem is still not solved. For my problem, $x$ has some relation with $a$, which is unknown until now. Thus, $a$ can not tend to $0$. Besides, I assume that $a\to 0$, the result is not very good.

Answer 1

You could use AsymptoticSolve. First, the zeroth order solution:

y0 = y /. First @ Solve[Log[b/y]==(x/y-a)^𝛽/.a->0, y]

Solve::ifun: Inverse functions are being used by Solve, so some solutions may not be found; use Reduce for complete solution information.

(-((x^𝛽 𝛽)/ProductLog[-b^-𝛽 x^𝛽 𝛽]))^(1/𝛽)

Then using AsymptoticSolve:

AsymptoticSolve[Log[b/y] == (x/y-a)^𝛽, {y, y0}, a->0]

{{y -> ConditionalExpression[(-((x^𝛽𝛽)/ProductLog[-((x^𝛽𝛽)/b^𝛽)]))^𝛽^(-1) + ((-Log[b/(-((x^𝛽𝛽)/ProductLog[-((x^𝛽𝛽)/b^𝛽)]))^𝛽^(-1)] + (x/(-((x^𝛽𝛽)/ProductLog[-((x^𝛽𝛽)/b^𝛽)]))^𝛽^(-1))^𝛽)* (-((x^𝛽𝛽)/ProductLog[-((x^𝛽𝛽)/b^𝛽)]))^𝛽^(-1))/ (-1 + 𝛽*(x/(-((x^𝛽𝛽)/ProductLog[-((x^𝛽𝛽)/b^𝛽)]))^𝛽^(-1))^𝛽) - (a𝛽(-1 - Log[b/(-((x^𝛽𝛽)/ProductLog[-((x^𝛽𝛽)/b^𝛽)]))^𝛽^(-1)] + 𝛽Log[b/(-((x^𝛽𝛽)/ProductLog[-((x^𝛽𝛽)/b^𝛽)]))^𝛽^(-1)] + (x/(-((x^𝛽𝛽)/ProductLog[-((x^𝛽𝛽)/b^𝛽)]))^𝛽^(-1))^𝛽) (x/(-((x^𝛽𝛽)/ProductLog[-((x^𝛽𝛽)/b^𝛽)]))^𝛽^(-1))^𝛽* (-((x^𝛽𝛽)/ProductLog[-((x^𝛽𝛽)/b^𝛽)]))^(2/𝛽))/ (x*(-1 + 𝛽*(x/(-((x^𝛽𝛽)/ProductLog[-((x^𝛽𝛽)/b^𝛽)]))^𝛽^(-1))^𝛽)^2), -Log[b/(-((x^𝛽𝛽)/ProductLog[-((x^𝛽𝛽)/b^𝛽)]))^𝛽^(-1)] + (x/(-((x^𝛽𝛽)/ProductLog[-((x^𝛽𝛽)/b^𝛽)]))^𝛽^(-1))^𝛽 == 0]}}