Graph of the function intersects the axis X at f = 0
so we need to solve the equation:
$$\frac{\log{\left(\cos{\left(2 x \right)} \right)}}{\log{\left(2 \right)}} = 0$$
Solve this equationThe points of intersection with the axis X:
Analytical solution$$x_{1} = 0$$
$$x_{2} = \pi$$
Numerical solution$$x_{1} = 94.2477796093519$$
$$x_{2} = 72.2566310277135$$
$$x_{3} = -97.389372502654$$
$$x_{4} = 100.530964736304$$
$$x_{5} = 34.5575189914319$$
$$x_{6} = 37.6991120767477$$
$$x_{7} = 56.5486675731909$$
$$x_{8} = 40.8407038692067$$
$$x_{9} = 28.2743338651142$$
$$x_{10} = -75.3982239198207$$
$$x_{11} = -43.9822971744191$$
$$x_{12} = -50.2654822535294$$
$$x_{13} = 18.8495552720944$$
$$x_{14} = 81.681409243074$$
$$x_{15} = 6.28318528407908$$
$$x_{16} = -65.9734457646558$$
$$x_{17} = 43.982297169579$$
$$x_{18} = -37.6991118776023$$
$$x_{19} = -9.42477817254169$$
$$x_{20} = -53.4070750099862$$
$$x_{21} = 12.5663707984054$$
$$x_{22} = -15.7079632968187$$
$$x_{23} = 34.5575194141501$$
$$x_{24} = -28.274333671219$$
$$x_{25} = -59.6902604582916$$
$$x_{26} = 62.8318524651379$$
$$x_{27} = 0$$
$$x_{28} = 81.6814092224531$$
$$x_{29} = 65.9734457532363$$
$$x_{30} = -75.3982236054042$$
$$x_{31} = -94.2477794177114$$
$$x_{32} = 21.9911485852348$$
$$x_{33} = 87.9645943363558$$
$$x_{34} = -97.3893721997423$$
$$x_{35} = 59.6902606605194$$
$$x_{36} = -6.28318508874543$$
$$x_{37} = -72.2566308356894$$
$$x_{38} = -31.4159267547793$$
$$x_{39} = 59.6902606597981$$
$$x_{40} = 15.7079634939052$$
$$x_{41} = -31.4159264120844$$
$$x_{42} = -53.4070753372009$$
$$x_{43} = -87.9645943581379$$
$$x_{44} = 50.2654824463146$$
$$x_{45} = 12.5663704095248$$
$$x_{46} = -21.9911485864121$$
$$x_{47} = -81.6814090389034$$
$$x_{48} = 78.5398161548118$$
$$x_{49} = 84.8230010599183$$
$$x_{50} = -9.42477780989129$$