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$$\frac{-1 + \frac{\log{\left(5 x \right)}}{1 - \log{\left(5 x \right)}^{2}}}{x^{2} \sqrt{1 - \log{\left(5 x \right)}^{2}}} = 0$$
Solve this equationThe roots of this equation
$$x_{1} = \frac{e^{- \frac{1}{2} + \frac{\sqrt{5}}{2}}}{5}$$
$$x_{2} = \frac{1}{5 e^{\frac{1}{2} + \frac{\sqrt{5}}{2}}}$$
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
$$\left[\frac{e^{- \frac{1}{2} + \frac{\sqrt{5}}{2}}}{5}, \infty\right)$$
Convex at the intervals
$$\left(-\infty, \frac{e^{- \frac{1}{2} + \frac{\sqrt{5}}{2}}}{5}\right]$$