Recover $f$ if we know that $frac{d}{dx} log f(x)$ and $f(x) to 0$ as $x to infty$

Let me set up the question by stating something that is well-known. Suppose $f$ is an unknown function, but we are given $f’$ and the fact that
begin{align}
lim_{x to infty} f(x)=0.
end{align}
Then, by using fundamental theorem of calculus we have that
begin{align}
f(a)-f(x)= int_x^a f'(t) dt
end{align}
and by using that $f(a) to 0$ as $ato infty$ we arrive at
begin{align}
f(x)= int_x^infty -f'(t) dt.
end{align}

In other words, we can recover $f$ based on this information.

Here is the question: Suppose instead we know $frac{d}{dx} log f(x)=frac{f'(x)}{f(x)}$ and $lim_{x to infty} f(x)=0$. Can we still recover $f(x)$?

If we use, the previous approach we arrive at
begin{align}
frac{f(x)}{f(a)}= e^{-int_x^a frac{f'(t)}{f(t)} dt}.
end{align}
Clearly, we have a problem with this approach as we get a division by zero in the limit.

My guess is that in general, it is not possible to recover $f(x)$ based on this information. (See one of the answers) However, can we do this with some extra minimal assumptions on $f$?

One trivial assumption that I would like to avoid making, is that we know $f(a)$ at some point $a$ which would make the FTC approach work.