Answer
True.
The proof is given below.
Work Step by Step
Make $x=\frac{a}{b}$ and use the One-to-One Property:
$\ln x=\ln\frac{a}{b}=ln a-\ln b$
If $f(x)\lt0$, then $ln a- \ln b\lt0~→~\ln a\lt\ln b$
We know that $f(x)=\ln x$ is ascending for all $x$ in the Domain: $(0,∞)$. So, if $\ln a\lt\ln b$ then $a\lt b~→~\frac{a}{b}\lt1$, that is, $x\lt1$.
