$f(x) = \log x$ for any real number $x > 0$ and
$g(n)=\begin{cases} n& \text{if $n$ is even}\\ \tfrac{1}{n}& \text{if $n$ is odd}.\end{cases}$
for any natural number $n$.
If $x$ is a natural number greater than $1$, then what is the value of
$f(f(x^{g(10)})) – f(f(x^{g(9)})) + f(f(x^{g(8)})) –\cdots – f(f(x^{g(1)}))$
What is the best way to this?