How to simplify the expression below

Question

I was stuck with simplifying this expression:

$$\sum_{i=1}^{2^{n}} \log (i) $$

I was told that it equals to $$n\times2^{n}$$ Thank you for your help!

Answer 1

Hint:

1. $\log(a) + \log(b) = \log(a\cdot b)$
2. $1\cdot 2\cdot\cdots\cdot (t - 1)\cdot t = t!$
3. Stirling's approximation: $\ln (r!) \simeq r\ln (r)$
4. $\log_c c = 1$