Existence of a monotonic-decreasing function with monotonically increasing log derivative

For reasons that are completely absurd, I’m interested in whether such a function $f$ (as briefly described in the title) exists:

$f$ is smooth on the interval $(0,1)$.

$f(0) = 1$

$f(1) = 0$

$f$ is monotonically decreasing on the interval $(0,1)$

$d/dx (ln f) < 0$ and is monotonically increasing on the interval $(0,1)$.

If possible, $d/dx (ln f) to 0$ as $x to 1$. But this condition is not a requirement.

I suspect that such a function does not exist, because as the function approaches zero, its logarithmic derivative will diverge – that is, unless it’s (standard) derivative goes as $f(x) g(x)$ where $g(x)$ goes to zero in the limit where $x to 1$.