Evaluate the product \(\prod_{k=1}^n 2 \cdot 4^k\).

This expression can be rearranged a few times by taking advantage of identities found in this appendix. We start by taking the log of the product to turn it into a summation:

\[\begin{split} & \lg \left( \prod\limits_{k=1}^n 2 \cdot 4^k \right) \\ &= \sum\limits^{n}_{k=1} \lg (2 \cdot 4^k)\\ &= \sum\limits^{n}_{k=1} \lg(2) + k \lg (4)\\ &= \left( \lg(2) \sum\limits^{n}_{k=1} 1 \right) + \left( \lg(4) \sum\limits^{n}_{k=1} \right) \\ &= n + 2\frac{n(n+1)}{2} \\ &= n(n+2) \end{split}\]

In order to unwind the applied log in the first step, we raise \(2\) to the power of our result:

\[2^{n(n+2)} = 2^{n^2} \cdot 4^n\]