A1 Determine the positive integer $n$ such that $8^{4}=4^{n}$.
Solution: The answer is $n=6$.

Solution 1: Note that $8^{4}=\left(2^{3}\right)^{4}=2^{12}=4^{6}$. Therefore, $n=6$.

Solution 2: We write $8^{4}$ and $4^{n}$ as an exponent with base 2 .
$$
\begin{array}{ll}
8^{4} & =4^{n} \\
\left(2^{3}\right)^{4} & =\left(2^{2}\right)^{n} \\
2^{12} & =2^{2 n}
\end{array}
$$

Therefore, $2 n=12$. Hence, $n=6$. $\square$
Solution 3: Note that $8^{4}=\left(8^{2}\right)^{2}=64^{2}=4096$. Hence, $4^{n}=4096$. We check each positive integer $n$ starting from 1 .

All positive integers $n>6$ yield a value of $4^{n}$ larger than 4096 . Therefore, $n=6$. $\square$