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Real Resolution II (Posted on 2014-11-18) Difficulty: 3 of 5
Determine the total number of real numbers that satisfy this equation:

log2(x/5) = sin(5*pi*x).

** All the angles are in radians.

No Solution Yet Submitted by K Sengupta    
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Solution solution Comment 1 of 1
Where log2(x/5) starts being larger than -1 is found by

x/5 = 2^-1 = .5
x = 5/2

Where that function goes above 1 is found by

x/5 = 2^1 = 2
x = 10

sin(5*pi*x) has a period of 2/5 = .4, so at x=2.5, the function has value 1 (it's at the crest of the wave), having gone through 6.25 cycles.

At x= 10, the sine wave will have gone through 25 complete cycles past the origin, or 18.75 cycles from where x = 2.5, when the log curve came above -1.

As the sine curve in this range starts at a peak, we can equally call it a cosine curve, which helps in picturing it.

Each of the first 18 complete eligible cycles contains two intersections of the two curves, as the cosine curve dips down and up again; the remaining 3/4 of a cycle contains the dip, but the upward rebound only reaches zero--too low to hit the log curve.

So:  2*18 + 1 = 37 is the answer.

Edited on November 18, 2014, 2:37 pm
  Posted by Charlie on 2014-11-18 14:36:03

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