1999 AHSME Problems/Problem 18
Contents
Problem
How many zeros does 
have on the interval 
?
Solution
For we have 
, and the logarithm is a strictly increasing function on this interval.
is zero for all 
of the form 
, where 
. There are 
such in 
.
Here's the graph of the function on 
:
![[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]](http://latex.artofproblemsolving.com/c/f/2/cf2431375372a86356755fa21f598ca5a41c2d7c.png)
As we go closer to 
, the function will more and more wildly oscilate between 
and 
. This is how it looks like at 
.
![[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]](http://latex.artofproblemsolving.com/c/2/5/c2598278cc59119edb942ef4a3b8c14f08f8ce70.png)
And one more zoom, at 
.
![[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]](http://latex.artofproblemsolving.com/a/e/d/aed9faa016e21438c773ea933f787f5e39ef94bd.png)
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 
.
-PhysicsMan
See also
| 1999 AHSME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 17 |
Followed by Problem 19 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 | ||
| All AHSME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
