Bounded ration of functions If $f(x)$ is a continuous function on $\mathbb R$, and $f$ is doesn't vanish on $\mathbb R$, does this imply that the function $\frac{f\,'(x)}{f(x)}$ be bounded on $\mathbb R$?
The function $1/f(x)$ will be bounded because $f$ doesn't vanish, and I guess that the derivative will reduce the growth of the function $f$, so that the ration will be bounded, is this explanation correct!?

In fact for any positive continuous function $g(x)$ on $\mathbb R$ and any $y_0 \ne 0$, the differential equation $\dfrac{dy}{dx} = g(x) y$ with initial condition  $y(0) = y_0$ has a 
unique solution $y = f(x) = y_0 \exp\left(\int_0^x g(t)\ dt\right)$, and this is a nonvanishing continuous function with $f'(x)/f(x) = g(x)$.    So $f'/f$ can do anything 
a continuous function can.
