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- anonymous

Expand the logarithm
Quesition about placement -
My question is do I need to add parenthesis to logarithm answers?

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- anonymous

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- anonymous

My book has these all over the place so I'm not sure.

- anonymous

I think you should add parenthesis
idk i might be wrong

- jdoe0001

the "2" is multiplying the logarithmic function, thus if the logarithm expands to the "sum" version, the "2" has to multiply the expanded logarithmic version then

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- anonymous

so then yes add the parenthesis?

- jdoe0001

\(\bf log_4(3xyz)^2\implies 2\left( log_43+log_4x+log_4y+log_4z \right)
\\ \quad \\
2log_43+2log_4x+2log_4y+2log_4z\)

- anonymous

okay you're confusing me you just wrote both down

- jdoe0001

hmmm nope, the last line is the parenthesized version, expanded

- anonymous

okay so I don't add parenthesis.

- jdoe0001

well.. distributing the "2" will be, expanding, thus

- anonymous

- anonymous

I'm confused now and idk what I'm doing

- jim_thompson5910

the first step is to pull down the exponent 2 using this rule
\[\Large \log_{b}(x^y) = y*\log_b(x)\]

- anonymous

right

- jim_thompson5910

after you pulled down the 2, you will have
\[\Large 2\log_{4}\left(3xyz\right)\]

- jim_thompson5910

then you'll use the rule log(x*y) = log(x) + log(y) to expand out log(3xyz)

- anonymous

yep

- jim_thompson5910

use the rule log(x*y) = log(x) + log(y) to get...
\[\Large 2\log_{4}\left(3xyz\right)\]
\[\Large 2[\log_{4}\left(3xyz\right)]\]
\[\Large 2[\log_{4}(3)+\log_{4}(xyz)]\]
\[\Large 2[\log_{4}(3)+\log_{4}(x)+\log_{4}(yz)]\]
\[\Large 2[\log_{4}(3)+\log_{4}(x)+\log_{4}(y)+\log_{4}(z)]\]

- jim_thompson5910

I used the rule in the smallest pieces possible. So it's a bit more dragged out than it has to be. You can just go from log(3xyz) to log(3)+log(x)+log(y)+log(z) in one step really

- jim_thompson5910

anyways, the point is that 2 on the outside is being multiplied by each term inside. You can leave the 2 out there like it is, but make sure you have surrounding parenthesis around those log terms
don't write \[\Large 2\log_{4}(3)+\log_{4}(x)+\log_{4}(y)+\log_{4}(z)\]
instead write \[\Large 2[\log_{4}(3)+\log_{4}(x)+\log_{4}(y)+\log_{4}(z)]\]
or write \[\Large 2(\log_{4}(3)+\log_{4}(x)+\log_{4}(y)+\log_{4}(z))\]

- anonymous

yeah I have options to choose from and the parenthesis around the 3,x,y,and z isn't an option it's just
2log4 3+log4 x+log4 y +log4 z or
2(log4 3+log4 x+log4 y +log4 z) so I feel like I will just go w/ the parenthesis.

- jim_thompson5910

then go with 2(log4 3+log4 x+log4 y +log4 z)

- anonymous

my book has examples written like yours so i think that's why I was confused. Thank you for helping me

- jim_thompson5910

as for what jdoe0001 did, he simply used the distribution rule to multiply the '2' by each term inside
eg: 2*(x+y) = 2*x + 2*y

- jim_thompson5910

I'm glad I could help clear things up

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