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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]


See also

1999 AHSME (ProblemsAnswer KeyResources)
Preceded by
Problem 17
Followed by
Problem 19
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All AHSME Problems and Solutions

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