# 1999 AHSME Problems/Problem 18

## Problem

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

## Solution

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 $(0,1)$: $[asy] import graph; size(250,200,IgnoreAspect); real f(real t) {return cos(log(t));} draw(graph(f,0.01,1),red,"\cos(\log(x))"); xaxis("x",BottomTop,LeftTicks); yaxis("y",LeftRight,RightTicks(trailingzero)); attach(legend(),truepoint(E),20E,UnFill); [/asy]$

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 $(0.0001,0.02)$. $[asy] import graph; size(250,200,IgnoreAspect); real f(real t) {return cos(log(t));} draw(graph(f,0.0001,0.02),red,"\cos(\log(x))"); xaxis("x",BottomTop,LeftTicks); yaxis("y",LeftRight,RightTicks(trailingzero)); attach(legend(),truepoint(E),20E,UnFill); [/asy]$

And one more zoom, at $(0.000001,0.0005)$. $[asy] import graph; size(250,200,IgnoreAspect); real f(real t) {return cos(log(t));} draw(graph(f,0.000001,0.0005),red,"\cos(\log(x))"); xaxis("x",BottomTop,LeftTicks); yaxis("y",LeftRight,RightTicks(trailingzero)); attach(legend(),truepoint(E),20E,UnFill); [/asy]$

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 