Zeros of a Shifted Log

Let $f(x)=|\lg x|-kx-2$. Determine, with proof, which of the following claims are true.

1. If $k=0$, then $f(x)$ has two zeros. 2. There exists $k<0$ such that $f(x)$ has exactly one zero. 3. There exists $k<0$ such that $f(x)$ has three zeros. 4. There exists $k>0$ such that $f(x)$ has three zeros.