Graphs !
Algebra
Level
5
The number of solutions of the equation . is :
This problem is a part of the set advanced is basic
The answer is 6.
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1 solution
If one plots both |ln|x|| and sin(pi*x) in the xy-plane, it becomes clear that the two curves will never intersect over x = (-inf, -e) U (-1/e, 0) U (0, 1/e) U (e, +inf) as the logarithmic function will exceed 1 in each of these intervals. Thus we will restrict ourselves to the closed intervals [-e, -1/e] and [1/e, e]. For the sine curve we observe that:
sin(pi*x) = negative in [-e, -2); (-1, -1/e]; (1, 2) and non-negative in [-2, -1]; [1/e, 1]; [2, e]
and the logarithmic curve we be strictly non-negative over [-e, 0) U (0, e]. The intersection points can only occur in the intervals [-2, -1]; [1/e, 1]; [2, e] with 2 points contained in each. This gives 6 total intersection points in all.
I'd be happy to post a plot picture upon request!
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