# 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 $(0.000001,0.0005)$.

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

## Solution

If cos(log(x)) = zero, then log(x) = π/2 + nπ.

If we consider the limiting case as x approaches zero, log(x) approaches negative infinity.

If we consider the other boundary, x equals 1 where log(x) equals zero.

Due to the intermediate value theorem log(x) must contain all negative real numbers, and thus an infinite number of solutions to log(x) = π/2 + nπ. This means that cos(x) is zero an infinite number of times giving $\boxed{\text{(D) infinitely\ many}}$.

-PhysicsMan