## anonymous 3 years ago Express the given quantity as a single logarithm. 1/5 ln(x + 2)^5 + 1/2 [ln x − ln(x^2 + 3x + 2)^2]

1. anonymous

$\frac{ 1 }{ 5 }\log(x+2)^5+\frac{ 1 }{ 2 }[{\ln x- \ln (x^2+3x+2)^2}]= \frac{ 1 }{ 5 }*5 \log(x+2)+\frac{ 1 }{2 }[\ln x- \ln(x^2+3x+2)^2]$ i am sorry to say, this can't be unified because ln and log have different bases

2. anonymous

All of them are base e, surely?

3. anonymous

4. anonymous

Are there any nice identities for$\log(a)^b$? Maybe I'm being stupid.

5. anonymous

Also- does anyone know how to have latex flow with your text rather than indent?

6. anonymous

no you are right ...and it can be simplified

7. anonymous

do you know the property of logarithm ? so like ln a + ln b = ? or ln a -ln b = ? or ln a^2 = ?

8. anonymous

you dont like cooperating nothing ? i like help you here ... come on !!!

9. anonymous

i'm confused

10. anonymous

why ? so you know that ln a + ln b = ln a*b yes ?

11. anonymous

yes

12. anonymous

so than do you know how many will be ln a^2 =

13. anonymous

or ln a - ln b =

14. anonymous

ln(a/b)

15. anonymous

yes and ln a^2 =

16. anonymous

2ln a

17. anonymous

yes right sure so than these property of logarithms can us in case of your exercise too ?

18. anonymous

(ln(x + 2) (1/2 ln x / 2ln(x^2 + 3x + 2))

19. anonymous

[ln(x + 2)] [1/2( ln x / 2ln(x^2 + 3x + 2))]

20. anonymous

so the first part is right but the secondly you need to separate 1/5 ln(x+2)^5 = 5/5 ln(x+2) = ln(x+2) this is right so in the second part there are 1/2 (ln x - ln(x^2 +3x +2)^2) = 1/2 (ln (x/(x^2 +3x +2)^2) = ln (x/(x^2 +3x +2)^2)^(1/2) do you understand till now ? can you continue it ?

21. anonymous

yess

22. anonymous

so than how will be ?

23. anonymous

[ln(x+2)] [ln (x/(x^2 +3x +2)^2)^(1/2)]

24. anonymous

why ? than you know that (a^2)^(1/2) =a^(2/2) =a so than what will be ?

25. anonymous

and you know that ln a +ln b = ln a*b

26. anonymous

[ln(x+2)] [ln (x/(x^2 +3x +2)]

27. anonymous

not is right please check it because there is just (x^2 +3x +2)^2 and x on numerator have not exponent 2 so what sign that not can be simplified by 1/2 OK ?

28. anonymous

do you understand it sure ?

29. anonymous

no

30. anonymous

so the first part is right sure will be ln(x+2) OK . in the second part there are 1/2 (ln x - ln(x^2 +3x +2)^2) = 1/2 (ln (x/(x^2 +3x +2)^2)) = ln (x^(1/2) /((x^2 +3x +2)^2)^1/2 = ln (x^1/2) /(x^2 +3x +2) so can you continue it now ?

31. anonymous

so than will result there ln (x+2) + ln (x^1/2) /(x^2 +3x +2) so from this can you continue it ?

32. anonymous

are you here ?

33. anonymous

so than will be using the property of logarithm log a +log b = log a*b so than will be ln ((x+2)*x^1/2 ) /(x^2 +3x +2)

34. anonymous

so x^2 +3x +2 how can you factoriz it ?

35. anonymous

do you know ?

36. anonymous

so x^2 +3x +2 = (x+2)(x+1) is right ?

37. anonymous

so than will be ln (x+2)x^(1/2) /(x+2)(x+1) so simplifie by (x+2) and will result ln x^1/2 /(x+1) so do you understand it now sure ?