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I came across a high school math question which asks to find how many real numbers $x$ satisfy the equation $$\cos(3\pi x) = \frac12\log_7 x$$

I have no clue how to solve it with both $\cos$ and $\log$ coexist in an equation.

I used an online program to plot both graphs and, in theory, I could count all the intersections exist between $-1 \le y \le +1$ (since that's the limit of $\cos (3\pi x)$). I trust there must be a smarter way to compute this?

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Samuel Wong
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    You are almost certainly meant to solve this problem with computer assistance, in the form of looking at a graph. Edit: upon further thought, there will be many zeros, and you'll need to use the periodicity of the cosine function to do some smart counting on the interval where $ 1/2 log_7(x) $ is between $ -1 $ and $ 1 $. – Jake Mirra Oct 17 '20 at 14:17

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I've highlighted the region between $ x = 1 $ and $ x = 48 $ where you do not need to count the roots - you can simply use the fact that sinusoid curve will pass through the logarithm curve twice per period. Since the period is $ P = 2/3 $, there will be approximately 141 roots in that interval. Near the boundary of the region I highlighted, you will need to use a graphing calculator to count the intersections. enter image description here

If we zoom in on the left boundary we see this: enter image description here

and if we zoom in on the right boundary we see this: enter image description here

By this reasoning I count 3 roots in [0,1], 141 roots in [1,48], and 3 roots in [48,49]. Beyond $ x = 49 $, the logarithm curve leaves the range of the sinusoid, so there will be no more roots. So by my count, I see 147 roots.

Jake Mirra
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