How many zeros does the function y = sin(lnx) have for 0<x<1 ?
$\displaystyle \sin(\ln|x|)=0$
$\displaystyle \ln|x| = \sinh(0) \Rightarrow \ldots -\pi, 0, \pi \ldots$
$\displaystyle x = e^{\pi n}$ for $\displaystyle n \ge 0$ where n is an integer and $\displaystyle 0 < x < 1$
Therefore $\displaystyle \pi n < 0$, so $\displaystyle n < 0$
$\displaystyle x = e^{\pi n} \forall n < 0, n \in Z$
$\displaystyle \forall$: for all* (isn't this supposed to have a horizontal line through the middle?)
However....