#### Answer

$$s=\int_{a}^{b}\sqrt{ \frac{f^2(x)+f'^2(x)}{f^2(x)} }$$

#### Work Step by Step

Since
$$y=\ln [f(x)]\to y'=\frac{f'(x)}{f(x)} $$
Then
\begin{align*}
s&=\int_{a}^{b}\sqrt{1+y'^2}dx\\
&= \int_{a}^{b}\sqrt{1+\frac{f'^2(x)}{f^2(x)} }\\
&=\int_{a}^{b}\sqrt{ \frac{f^2(x)+f'^2(x)}{f^2(x)} }\\
\end{align*}