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$$4^{- x} \left(\log{\left(4 \right)}^{2} \log{\left(x - 3 \right)}^{2} - \frac{10 \log{\left(4 \right)} \log{\left(x - 3 \right)}}{x - 3} - \frac{5 \left(\log{\left(x - 3 \right)} - 4\right)}{\left(x - 3\right)^{2}}\right) \log{\left(x - 3 \right)}^{3} = 0$$
Solve this equationThe roots of this equation
$$x_{1} = 113.444197021989$$
$$x_{2} = 41.9247540783901$$
$$x_{3} = 85.5031137944288$$
$$x_{4} = 83.5094220038088$$
$$x_{5} = 59.637391808103$$
$$x_{6} = 63.6059563990726$$
$$x_{7} = 7.24159390643843$$
$$x_{8} = 103.460476300814$$
$$x_{9} = 36.1648934309566$$
$$x_{10} = 38.0672006610478$$
$$x_{11} = 87.4971955595694$$
$$x_{12} = 77.5310997169679$$
$$x_{13} = 32.4566338341999$$
$$x_{14} = 55.6760450934624$$
$$x_{15} = 4$$
$$x_{16} = 51.7246263439374$$
$$x_{17} = 105.45689837203$$
$$x_{18} = 111.4471524729$$
$$x_{19} = 67.5799261737493$$
$$x_{20} = 5.06044220693169$$
$$x_{21} = 79.5233680889575$$
$$x_{22} = 47.7873726184949$$
$$x_{23} = 115.441372400767$$
$$x_{24} = 61.6209012458532$$
$$x_{25} = 27.6679929981534$$
$$x_{26} = 49.7538770181718$$
$$x_{27} = 81.5161588248275$$
$$x_{28} = 29.0420218596767$$
$$x_{29} = 43.8712946654158$$
$$x_{30} = 39.9888920422188$$
$$x_{31} = 65.5923544330541$$
$$x_{32} = 45.8260828406174$$
$$x_{33} = 95.476793131649$$
$$x_{34} = 89.4916331142587$$
$$x_{35} = 91.4863961066935$$
$$x_{36} = 71.5580438677232$$
$$x_{37} = 53.6988760653224$$
$$x_{38} = 73.5483669468169$$
$$x_{39} = 30.6894678200551$$
$$x_{40} = 57.6556733311909$$
$$x_{41} = 99.4682026938527$$
$$x_{42} = 69.5685294346847$$
$$x_{43} = 97.4723813501921$$
$$x_{44} = 75.5394108044439$$
$$x_{45} = 107.453492815044$$
$$x_{46} = 109.450247782127$$
$$x_{47} = 93.4814574993208$$
$$x_{48} = 101.464239625316$$
$$x_{49} = 34.2901447464208$$
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
$$\left[4, 5.06044220693169\right] \cup \left[7.24159390643843, \infty\right)$$
Convex at the intervals
$$\left(-\infty, 4\right] \cup \left[5.06044220693169, 7.24159390643843\right]$$