13. Given the function $f(x)=\log _{2}(x+1)$, and when the point $(x, y)$ moves on the graph of $f(x)$, the point $\left(\frac{x}{3}, \frac{y}{2}\right)$ moves on the graph of $y=g(x)$, find the maximum value of the function $h(x)=g(x)-$ $f(x)$.
13. $g\left(\frac{x}{3}\right)=\frac{1}{2} y=\frac{1}{2} \log _{2}(x+1)$, so $g(x)=\frac{1}{2} \log _{2}(3 x+1), h(x)=g(x)-$
$$
\begin{array}{l}
f(x)=\frac{1}{2} \log _{2}(3 x+1)-\log _{2}(x+1)=\frac{1}{2} \log _{2} \frac{3 x+1}{(x+1)^{2}}=\frac{1}{2} \log _{2}\left[\frac{3}{x+1}-\frac{2}{(x+1)^{2}}\right]= \\
\frac{1}{2} \log _{2}\left[-2\left(\frac{1}{x+1}-\frac{3}{4}\right)^{2}+\frac{9}{8}\right] .
\end{array}
$$