Henning has already pointed out in a comment that the question doesn't make sense when taken literally. I'll assume that you mean that $\ln |f|$ (note the absolute value) is concave wherever it is defined.
You seem to be assuming that $f$ is differentiable. (In fact the tags seem to indicate that $f$ is supposed to be a polynomial, but this isn't mentioned in the question, so I won't assume it.)
Now if $f$ is differentiable, then quite independent of the behaviour of $\ln |f|$, there has to be an extremum of $f$, and hence a root of $f'$, between any two roots of $f$. Thus there can't be more than one root between two stationary points, and the question is only whether there can be more than one stationary point between two roots.
Indeed your argument shows that this can't happen when $\ln |f|$ is concave, and this is a special case of the fact that log-concavity of a positive differentiable function implies unimodality, and this is a useful geometric interpretation of log-concavity to keep in mind. You can see this more directly by noting that where $f$ has an extremum, $\ln |f|$ has an extremum of the corresponding kind, and since a concave function can't have minima, a positive differentiable log-concave function can't have minima, and thus it can have at most one maximum.
For some applications of this connection in the case of sequences, see generatingfunctionology by Herbert Wilf.