Mixed up functions

I recently came across an AIME problem that compared what I call “mixed up functions:”

How many real numbers x satisfy the equation:

\frac{1}{5}(\log_2 x)  =sin(5\pi x)

By mixed up, I mean we have the periodic sine curve equated with the monotonically increasing log function.  Often with logs we can replace them with an exponential function, in this case with base 2, but that looks messy.

The approach I often see with these mis-matched functions is to draw a graph and see where the 2 functions intersect.  But first, I’m going to multiply both sides by 5 because who likes fractions?

\log_2 x  =5 sin(5\pi x)

Ah that’s better.  Notice that the maximum and minimum values of y=5 sin(5\pi x) occur when x=\frac{1}{10} and x=\frac{-1}{10}, and the maximum and minimum values are 5 and -5.  And of course this is a periodic function that is defined for all real numbers.

y=\log_2 x on the other hand is defined only over positive reals, and since it is always increasing from left to right, at some point the value of the function will exceed 5.  Specifically, this will occur for x>32.  Likewise, y=\log_2 x will be less than -5 for x<\frac{1}{32}.

And it’s apparent that in the interval \frac{1}{32}<x<32 the log function is going to intersect the sine function as it goes up and down on its periodic path.  How many times does it intersect?  AIME writers like to catch students on “off by one” errors, so take care to examine what is happening close to x=\frac{1}{32} and x=32 before entering your answer!




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