Hard trig problem, and Geometry probability problem:

How man solutions are there to the equation: log_10 x = sin(2x*pi) ?

The answer is 19, but I don't know how to do it.
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Question 2:

Suppose three points are independently chosen at random on the perimeter of a circle. What is the probability that all three lie in some semicircle? [that is what is the probability that there is a linen passing thru the center of the circle such that all the points are on one side of that line]

the answer is 3/4, but I don't know how to get it.
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Thanks for your help!

http://www.wolframalpha.com/input/?i=ln%…

they are numeric solutions
log x = sin(2πx)
x = 10^(sin2πx)

ln(x) = sin(2πx) ln(10)
ln(x) = ln(10) sin(2πx)

let left-hand side be:
y = ln(x)

let right-hand side be:
y = ln(10) sin(2πx)
the ln(10) is the amplitude and will determine the maximum value of y = ln(10) sin(2πx)

the two curves y = ln(10) sin(2πx) and y = ln(x) will have solutions where y = ln(x) is within the ln(10) amplitude
withing 0 ≤ x ≤ 10, there are 19 intersections.


ii)
n independent points

P(all n in some semicircle ) = n(½)^(n - 1)
n = 3 points

P(all 3 in some semicircle ) = 3(½)^(3 - 1) = ¾