We are tasked with solving the equation $\log_4(32 + 4^x)$.

Let's break it down step by step:

1. Rewrite the logarithmic equation: The logarithmic equation $\log_b(a) = c$ means $a = b^c$. In this case: $\log_4(32 + 4^x) = y$ implies: $32 + 4^x = 4^y$

2. Express $4^x$ and $4^y$: Both $4^x$ and $4^y$ can be expressed as powers of 2. Recall that $4 = 2^2$, so: $4^x = (2^2)^x = 2^{2x}$ and $4^y = (2^2)^y = 2^{2y}$ Thus, the equation becomes: $32 + 2^{2x} = 2^{2y}$

3. Simplify 32: Since $32 = 2^5$, we now have: $2^5 + 2^{2x} = 2^{2y}$

   At this point, we have to solve this equation for $x$.

Let me know if you would like to proceed further with solving this equation step by step!