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$$\frac{- \log{\left(x \right)} + 1 - \frac{\log{\left(x \right)}^{2}}{\log{\left(x \right)}^{2} - 1}}{x^{2} \sqrt{\log{\left(x \right)}^{2} - 1}} = 0$$
Solve this equationThe roots of this equation
$$x_{1} = e^{- \frac{\sqrt[3]{\frac{3 \sqrt{69}}{2} + \frac{27}{2}}}{3} - \frac{1}{\sqrt[3]{\frac{3 \sqrt{69}}{2} + \frac{27}{2}}}}$$
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
$$\left(-\infty, e^{- \frac{\sqrt[3]{\frac{3 \sqrt{69}}{2} + \frac{27}{2}}}{3} - \frac{1}{\sqrt[3]{\frac{3 \sqrt{69}}{2} + \frac{27}{2}}}}\right]$$
Convex at the intervals
$$\left[e^{- \frac{\sqrt[3]{\frac{3 \sqrt{69}}{2} + \frac{27}{2}}}{3} - \frac{1}{\sqrt[3]{\frac{3 \sqrt{69}}{2} + \frac{27}{2}}}}, \infty\right)$$