Let $f(x)=\log (4 x), g(x)=\log (2 x)$ , And $h(x)=\frac{f(x)-g(x)}{f(x)+g(x)}$ , With $x>0$

Let $f(x) =\log (4 x) , g(x) =\log (2 x)$ , and $h(x) =\frac{f(x) -g(x) }{f(x) +g(x) }$ , with $x>0$ . Which of the following is a simplified formula for $h(x)$ ?

A) $\frac{\log (2 x) }{\log (6 x) }$
B) $\log \left(\frac{1}{3}\right)$
C) $2 \log (4 x) +2 \log (2 x)$
D) $\frac{\log (2) }{\log \left(8 x^{2}\right) }$