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$$\frac{\sin{\left(\log{\left(x \right)} \right)} - 3 \cos{\left(\log{\left(x \right)} \right)}}{x^{3}} = 0$$
Solve this equationThe roots of this equation
$$x_{1} = e^{\operatorname{atan}{\left(3 \right)}}$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 0$$
$$\lim_{x \to 0^-}\left(\frac{\sin{\left(\log{\left(x \right)} \right)} - 3 \cos{\left(\log{\left(x \right)} \right)}}{x^{3}}\right) = - \infty \operatorname{sign}{\left(\left\langle -4, 4\right\rangle \right)}$$
Let's take the limit$$\lim_{x \to 0^+}\left(\frac{\sin{\left(\log{\left(x \right)} \right)} - 3 \cos{\left(\log{\left(x \right)} \right)}}{x^{3}}\right) = \infty \operatorname{sign}{\left(\left\langle -4, 4\right\rangle \right)}$$
Let's take the limit- the limits are not equal, so
$$x_{1} = 0$$
- is an inflection point
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
$$\left[e^{\operatorname{atan}{\left(3 \right)}}, \infty\right)$$
Convex at the intervals
$$\left(-\infty, e^{\operatorname{atan}{\left(3 \right)}}\right]$$