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I am trying to simplify the following term: $$2^{\sum_{i=1}^n k \,log(i)}$$ So far I have only been able to come up with the following step: $$2^{\sum_{i=1}^n k \,log(i)}$$ $$= 2^{\sum_{i=1}^n \,log(i^k)}$$ $$= 2^{\sum_{i=1}^n \,log(i^k)}$$

What else can I do to simplify this term even further?

phoebe_albeduddel
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    Hello and welcome to math.stackexchange. Try to solve a simpler version at first. E.g.., consider the problem with $k = 1$ Perhaps work it out explicitly for $n = 3$ or $n = 4$. You may even try to pick an particular logarithm, e.g. $\log_2$, and then try to generalize. – Hans Engler Apr 22 '22 at 15:22
  • Thanks, using this approach and @Deepak's comment I was able to solve it. – phoebe_albeduddel Apr 24 '22 at 13:52

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Hints: $\log a + \log b = \log ab$

Also, $1\cdot 2\cdot 3 \dots n = n!$

Deepak
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