1. **State the problem:** Find the value of $\log_4 64$. 2. **Recall the definition of logarithm:** $$\log_b a = c \iff b^c = a$$ This means the logarithm answers the question: "To what power must the base $b$ be raised, to get $a$?" 3. **Apply the definition:** We want to find $c$ such that: $$4^c = 64$$ 4. **Express both numbers as powers of the same base:** Note that $4 = 2^2$ and $64 = 2^6$. So: $$ (2^2)^c = 2^6 $$ 5. **Simplify the left side using power of a power rule:** $$ 2^{2c} = 2^6 $$ 6. **Since the bases are equal, set the exponents equal:** $$ 2c = 6 $$ 7. **Solve for $c$:** $$ c = \frac{6}{2} = 3 $$ **Final answer:** $$\log_4 64 = 3$$