Let's find the inflection points, we'll need to solve the equation for this
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
the second derivative$$\frac{x^{\frac{\log{\left(x - 2 \right)}}{\log{\left(4 \right)}}} \left(\frac{\left(\frac{\log{\left(x \right)}}{x - 2} + \frac{\log{\left(x - 2 \right)}}{x}\right)^{2}}{\log{\left(4 \right)}} - \frac{\log{\left(x \right)}}{\left(x - 2\right)^{2}} + \frac{2}{x \left(x - 2\right)} - \frac{\log{\left(x - 2 \right)}}{x^{2}}\right)}{\log{\left(4 \right)}} = 0$$
Solve this equationThe roots of this equation
$$x_{1} = 2.55757872291725$$
Сonvexity and concavity intervals:Let’s find the intervals where the function is convex or concave, for this look at the behaviour of the function at the inflection points:
Concave at the intervals
$$\left[2.55757872291725, \infty\right)$$
Convex at the intervals
$$\left(-\infty, 2.55757872291725\right]$$