how can I solve this logarithmic equation?

Question

I've tried NSolve, Solve and Reduce for solving this equation but non of them worked. would you please help me?

NSolve[Log[Sqrt[x]/(1 + Sqrt[x])] == 3 Sqrt[1 - x], x]
NSolve::nsmet: This system cannot be solved with the methods available to NSolve.

Answer 1

Divide by "Sqrt[1-x]" and define the function:

f[x_] = 1/Sqrt[1 - x] Log[Sqrt[x]/(1 + Sqrt[x])]

Now check the behavior of this function. It will be concentrated around 0 and 1. Note the phase as indicated by the legend:

ComplexPlot3D[f[x], {x, -3 - 3 I, 3 + 3 I}, PlotRange -> {0, 4}, 
 PlotLegends -> Automatic]

You see that the function comes nowhere near +3. It would be red and have a magnitude of 3. Therefore, your equation has no solution in the complex numbers.

Answer 2

You can try fixed point iteration:

(* First find a suitable rearrangement by replacing an x with c and solve *)
eqn = Log[Sqrt[x]/(1 + Sqrt[x])] == 3 Sqrt[1 - c];

(* 1/9 (9 - Log[Sqrt[x]/(1 + Sqrt[x])]^2) *)

(* Now find a fixed point if one exists *)
FixedPoint[1 - (Log[Sqrt[#]/(1 + Sqrt[#])]/3)^2 &, 1.0, 50]

(* 0.9443735327111197` *)

(* Try it out on both sides: *)
Log[Sqrt[0.9443735327111197`]/(1 + Sqrt[0.9443735327111197`])]
(* -0.707558 *)

3 Sqrt[1 - 0.9443735327111197`]
(* 0.707558 *)

So as you can see, @josh 's comment about the branches is right, as the Sqrt on the right hand side is using the negative square root. This agrees with FindInstance if you square both sides:

x /. First@FindInstance[9 (1 - x) == Log[1 - 1/(1 + Sqrt[x])]^2, x]
(* Root[{9 - Log[1 - (1 + #^Rational[1, 2])^(-1)]^2 - 9 #& , 
  0.9443735327111196808405153254286572521300613203544128514706`30.15051499783199}] *)