Graph of the function intersects the axis X at f = 0
so we need to solve the equation:
$$\log{\left(2 \cos{\left(x \right)} \right)} = 0$$
Solve this equationThe points of intersection with the axis X:
Analytical solution$$x_{1} = \frac{\pi}{3}$$
$$x_{2} = \frac{5 \pi}{3}$$
Numerical solution$$x_{1} = -19.8967534727354$$
$$x_{2} = 38.7463093942741$$
$$x_{3} = -233.525053916841$$
$$x_{4} = 49.2182849062401$$
$$x_{5} = -86.9173967493176$$
$$x_{6} = 93.2005820564972$$
$$x_{7} = -51.3126800086333$$
$$x_{8} = 63.8790506229925$$
$$x_{9} = 5.23598775598299$$
$$x_{10} = 45.0294947014537$$
$$x_{11} = -61.7846555205993$$
$$x_{12} = -74.3510261349584$$
$$x_{13} = 30.3687289847013$$
$$x_{14} = -55.5014702134197$$
$$x_{15} = 26.1799387799149$$
$$x_{16} = 74.3510261349584$$
$$x_{17} = -24.0855436775217$$
$$x_{18} = -70.162235930172$$
$$x_{19} = -26.1799387799149$$
$$x_{20} = 13.6135681655558$$
$$x_{21} = -82.7286065445312$$
$$x_{22} = 32.4631240870945$$
$$x_{23} = -95.2949771588904$$
$$x_{24} = 89.0117918517108$$
$$x_{25} = 101.57816246607$$
$$x_{26} = -5.23598775598299$$
$$x_{27} = -80.634211442138$$
$$x_{28} = -17.8023583703422$$
$$x_{29} = -99.4837673636768$$
$$x_{30} = 57.5958653158129$$
$$x_{31} = 55.5014702134197$$
$$x_{32} = -68.0678408277789$$
$$x_{33} = 76.4454212373516$$
$$x_{34} = 68.0678408277789$$
$$x_{35} = 24.0855436775217$$
$$x_{36} = 86.9173967493176$$
$$x_{37} = 42.9350995990605$$
$$x_{38} = -93.2005820564972$$
$$x_{39} = -63.8790506229925$$
$$x_{40} = -11.5191730631626$$
$$x_{41} = 36.6519142918809$$
$$x_{42} = -76.4454212373516$$
$$x_{43} = 11.5191730631626$$
$$x_{44} = 70.162235930172$$
$$x_{45} = -49.2182849062401$$
$$x_{46} = -57.5958653158129$$
$$x_{47} = 61.7846555205993$$
$$x_{48} = -32.4631240870945$$
$$x_{49} = 17.8023583703422$$
$$x_{50} = 19.8967534727354$$
$$x_{51} = 80.634211442138$$
$$x_{52} = 82.7286065445312$$
$$x_{53} = -7.33038285837618$$
$$x_{54} = -101.57816246607$$
$$x_{55} = 99.4837673636768$$
$$x_{56} = -13.6135681655558$$
$$x_{57} = -30.3687289847013$$