I notice that the equation $x^{2y} = 4$ involves exponents, so I might want to try taking the logarithm of both sides to simplify it. If I take the logarithm base 4 of both sides, I get $\log_4(x^{2y}) = \log_4(4)$. Using the power rule of logarithms, I can bring down the exponent and write $2y \log_4(x) = \log_4(4)$. I also know that $\log_4(4) = 1$, since 4 is the base of the logarithm. So I have $2y \log_4(x) = 1$. Now I can use the given value of $x = 4$ and plug it into the equation. I get $2y \log_4(4) = 1$. But $\log_4(4) = 1$, so I can simplify this further and write $2y = 1$. Dividing both sides by 2, I get $y = \frac{1}{2}$. 
If $x^{2y}= 4$ and $x = 4$, what is the value of $y$? Express your answer as a common fraction.