Lets use this property -
log_2 x^n = n log_2 x
So log_2 x + log_2 sqrt(x) + log_2 sqrt(sqrt(x)) + log_2 sqrt(sqrt(sqrt(x))) .....= 4
=> log_2 x + 1/2 log_2 x + 1/4 log_2 x + 1/8 log_2 x .....= 4
=> log_2 x [1 + 1/2 + 1/4 + 1/8 ....] = 4
=> log_2 x [1/(1-1/2)] = 4
=> 2 log_2 x = 4
=> log_2 x = 2
=> x = 2^2
=> x = 4
Answer: 4