How many zeros does $f(x) = \cos(\log x)$ have on the interval $0 < x < 1$?
$\mathrm{(A) \ } 0 \qquad \mathrm{(B) \ } 1 \qquad \mathrm{(C) \ } 2 \qquad \mathrm{(D) \ } 10 \qquad \mathrm{(E) \ } \text{infinitely\ many}$
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 $(0.000001,0.0005)$.