How to solve this logarithm equation? How can I solve it?
$$ \frac{2}{\log_{8}(x-1)} - \frac{2}{\log_{8}(x )} =1$$  
I don't have idea how to solve it and I will be happy for help about this exercise.
 A: This problem does not have a closed-form solution.
Combine the fractions:
$$
\frac{2\log_8 x - 2\log_8 (x-1)}{\log_8 x \log_8 (x-1)}  =1 \\
\log_8 x - \log_8 (x-1) = \frac12 \log_8 x \log_8 (x-1)
$$
Then use properties of the logarithm:
$$
\log_8\left( \frac{x}{x-1} \right) = \log_8 \sqrt{ (x-1)^{\log_8 x }}
$$
Now raise $8$ to both sides of this equation and square both sides:
$$
\frac{x^2}{(x-1)^2} = (x-1)^{\log_8 x}$$
Handle the denominator on the left by adding to the exponent on the right:
$$x^2 = (x-1)^{2+\log_8 x}$$
This has a unique real positive solution at roughly $x = 3.7093175$, but there is no solution in closed form using only elementary functions. 
A: One can shift the equation by using:
\begin{align}
\log_{b}(x) &= \frac{\log_{d}(x)}{\log_{d}(b)} \\
x &\to t + \frac{1}{2}
\end{align}
to obtain
$$\frac{1}{\ln\left(t - \frac{1}{2}\right)} - \frac{1}{\ln\left(t + \frac{1}{2}\right)} = \frac{1}{6 \, \ln(2)}.$$
The solution for $t$ is $t \approx  3.20931751$ and yields $x \approx  3.70931751$.
