## anonymous one year ago Expand the logarithm Quesition about placement - My question is do I need to add parenthesis to logarithm answers?

1. anonymous

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

2. anonymous

I think you should add parenthesis idk i might be wrong

3. 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

4. anonymous

so then yes add the parenthesis?

5. jdoe0001

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

6. anonymous

okay you're confusing me you just wrote both down

7. jdoe0001

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

8. anonymous

okay so I don't add parenthesis.

9. jdoe0001

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

10. anonymous

@jim_thompson5910

11. anonymous

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

12. jim_thompson5910

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

13. anonymous

right

14. jim_thompson5910

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

15. jim_thompson5910

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

16. anonymous

yep

17. jim_thompson5910

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

18. 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

19. 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 $\Large 2(\log_{4}(3)+\log_{4}(x)+\log_{4}(y)+\log_{4}(z))$

20. 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.

21. jim_thompson5910

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

22. anonymous

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

23. 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

24. jim_thompson5910

I'm glad I could help clear things up