ttyme411gl

2022-07-10

What's the formula to solve summation of logarithms?
I'm studying summation. Everything I know so far is that:

Unfortunately, I can't find neither on my book nor on the internet what the result of:
$\sum _{i=1}^{n}\mathrm{log}i$
$\sum _{i=1}^{n}\mathrm{ln}i$
is.
Can you help me out?

Alexis Fields

Expert

By using the fact that
$\mathrm{log}a+\mathrm{log}b=\mathrm{log}ab$
then
$\sum ^{n}\mathrm{log}i=\mathrm{log}\left(n!\right)$
$\sum ^{n}\mathrm{ln}i=\mathrm{ln}\left(n!\right)$

Ciara Mcdaniel

Expert

Since $\mathrm{log}\left(A\right)+\mathrm{log}\left(B\right)=\mathrm{log}\left(AB\right)$, then $\sum _{i=1}^{n}\mathrm{log}\left(i\right)=\mathrm{log}\left(n!\right)$. I'm not sure if this helps a lot since you have changed a summation of $n$ terms into a product of $n$ factors, but it's something.