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My question is simple (but difficult for me):

$\prod(x)$ be expressed interms of $\sum (x)$

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The usual way to turn products into sums is to use logarithms. That's largely what they are for! –  Unwisdom Mar 11 at 18:23
    
@Unwisdom: Sums into products! –  Nick Mar 11 at 18:24
    
Sorry. But to go the other way, well, take a guess! –  Unwisdom Mar 11 at 18:32
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I think the asker is looking for something along these lines: math.stackexchange.com/a/708331/8530 –  Mats Granvik Mar 11 at 18:37
    
@MatsGranvik: Does this answer help in whatever you were trying to do? –  Nick Mar 11 at 18:58

1 Answer 1

up vote 2 down vote accepted

$$\prod_k x_k = \exp\left( \sum_k \ln x_k \right),$$ and as Unwisdom wrote in the comments, if you want to go the other way, $$\sum_k x_k = \ln \left( \prod_k \exp(x_k) \right).$$

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Could you please explain your answer. A formal proof, perhaps? Or atleast links –  Nick Mar 11 at 19:22
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@Nick For finite sums/products, it is just a consequence of these equations: $$\ln(ab)=\ln(a)+\ln(b) \quad \exp(a+b)=\exp(a)\exp(b).$$ For infinite sums and products, you have to take limits and use the fact that both exponentials and logarithms are continuous, but the principle is the same. –  Unwisdom Mar 11 at 20:06

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