Expand the function in a Maclaurin series $\ln(5\cos^{3}(x))$ $$\ln(5\cos^{3}(x))$$ Need to be expanded: $x^{4}$
I need to end this problem.
So I laid the beginning of the function.
$$\cos x=1-\frac{x^2}{2!}-\frac{x^4}{4!}+o(x^4)$$
$$5\cos^3(x)=5\left(1-\frac{x^2}{2!}+o(x^4)\right)^3=5-\frac{15x^2}{2!}+\frac{35}{8}x^4+o(x^4).$$
I do not know what to do next.

Since 
$$\cos x = \prod_{n\geq 0}\left(1-\frac{4x^2}{(2n+1)^2 \pi^2}\right)$$
we have:
$$\log \cos x = -\sum_{n \geq 0}\sum_{m\geq 1}\frac{4^m x^{2m}}{m(2n+1)^{2m}\pi^{2m}}=-\sum_{m\geq 1}\frac{(4^m-1)\,\zeta(2m)\,x^{2m}}{m\pi^{2m}}$$
so, in a neighbourhood of $x=0$:
$$\log(5\cos^3 x) = \log 5-3\sum_{m\geq 1}\frac{(4^m-1)\,\zeta(2m)\,x^{2m}}{m\pi^{2m}}=\log 5-\frac{3}{2}x^2-\frac{1}{4}x^4+O(x^6).$$
