Analytical value of minima for a function involving Logarithm and exponential

Question

I am trying to derive the analytical expression for the point of minima of the function $$\frac{c+ x \ln(-2+2 e^{c/x})-\ln (-1+e^{2c/x})}{x [\ln(-1+e^{c/x})-\ln (-1+e^{2c/x}) ]} $$ Here $x$ is the variable and $c$ is a constant. Plot of the function for a specific $c$ value is shown in the figure: Mathematica code:

(c + x (Log[-2 + 2 E^(c/x)] - 
Log[-1 + E^((2 c)/x)]))/(x (Log[-1 + E^(c/x)] - 
Log[-1 + E^((2 c)/x)]))

Any help will be appreciated. I have also asked this question on mathematica stack exchange.

Answer 1

Making $y = e^{\frac cx}$ after some simplifications like $\ln(y^2-1) = \ln(y-1)+\ln(y+1)$ we have

$$ f(y) = 1-\frac{\ln(2 y)}{\ln(y+1)} $$

with graphics shown in blue. Now the minimum for $f(y)$ (using an iterative method) is attained at $y^*=e^{\frac {c}{x^*}} \approx 3.4034978790622934$

then

$$ x^* \approx \frac{c}{\ln(3.4034978790622934)} $$