What do you mean by

output involving the ProductLog function, but how to extract something from it

The ProductLog function accepts a and b as parameters. What do you want to "extract"?

Does this help? Have a look at xx[[2]] as well. 3 and 4 give essentially zero.

tt = ((pi*x - (pi/2 b)*a*x + pi*(a/x)*Log[1 + (1/2 b)*x^2]) x // FullSimplify)
xx = Solve[tt == 0, x] // Flatten
fx = x /. xx[[1]]
Plot3D[{Re[fx], Im[fx]}, {a, .1, 1}, {b, .1, 1},  PlotStyle -> {{Opacity[.5], Blue}, {Opacity[.4], Red}}]

Thanks Hans for your reply and the lines of code provided. Essentially through your approach I end up with the same answer that I got previously, if I'm not letting anything go unnoticed. What I would like to know if it's possible to get symbolical solutions for x depending on the parameters "a" and "b". My ultimate goal is to obtain a phase diagram depending on "a" and "b". Therefore, later on, I would have to calculate second derivatives and evaluate them in the extremes candidates obtained. In all attempts, I always have an output involving the ProductLog function, but how to extract something from it, that's where I get stuck.