smc

For $0<x<1$ we have $-\infty <\log x<0$, and the logarithm is a strictly increasing function on this interval. $\cos (t)$ is zero for all $t$ of the form $\frac{\pi }{2}+k\pi$, where $k\in \mathbb{Z}$. There are $\boxed{\text{infinitely many}}$ such $t$ in $(-\infty ,0)$. Here's the graph of the function on $(0,1)$: As we go closer to $0$, the function will more and more wildly oscilate between $-1$ and $1$. This is how it looks like at $(0.0001,0.02)$. And one more zoom, at $(0.000001,0.0005)$.