What is the best method for fitting a curve that has the dependent variable on both sides of the equation? I am trying to fit a curve to a set of measured data. Similar studies have been done, and the resulting curve fit is usually of the following form.
$$\frac{1}{\sqrt{Y}}=a \log{\left(X \sqrt{Y}\right)}-b$$
I need to find the $a$ and $b$ that will result in the least error; I think maximizing $R^2$ would suffice.
The method to do this is not clear to me, as $Y$ is found on both sides of the equation with no obvious way to simplify. Is iteratively solving a system of equations (left and right sides) my best option?
I currently have the data in excel, which provided me a power fit that is not sufficiently accurate. Equations of the form above have a more accurate trend. I plan to do this new curve fitting in python, as it would probably be a pain in excel.
Ideally I would simplify this equation to be of the form $Y=f(X)$, as it is the $Y$ that I need to calculate from the $X$ when using my model. I just don't know a good way to do that.

For each couple $(X,Y)$ compute $x=\log( X \times \sqrt{Y} )$ and $y=1/\sqrt Y.$
Then the data is the couple $(x,y)$ related as $y=a \cdot x - b. $
$a$ and $b$ are obtained thanks to linear regression.
