Domainomania

Before you start, keep following properties in your mind: 1. log cannot have negative values. 2. The base of the log cannot be negative. 3. Cosine is only positive in first and fourth quadrant.

Now, for the first quadrant, cos is +ve for 0 to pie/2. Therefore, value of x varies from 0 to 1/4. Therefore, first range is (0,1/4). Similarly, there are two ways you can reach the last quadrant. The first range for x is (-1/4,0) and the second range is (3/4,1). Now, since x cannot be negative therefore, the (3/4,1) is the accepted range. Now, on comparing, the values, a=0 y=1 z=4 b=3 c=4 d=1, so on adding these we get 13.

For the function to be defined

$\log_x \cos2 \pi x \geq 0$

Case 1: When $x>1$

$\cos2 \pi x \geq x^{0}$

$\cos2 \pi x \geq 1$

$\cos2 \pi x = 1$

$x = {1,2,3,4,...}$

Case 2: When $0 < x < 1$

$0 < \cos2 \pi x < 1$

$0 < 2 \pi x < \dfrac{\pi}{2} or \dfrac{3 \pi}{2} < 2 \pi x < 2 \pi$

$0<x<\dfrac{1}{4} or \dfrac{3}{4}<x<1$

Required value = $0+1+4+3+4+1 = \boxed{13}$