Question

Express the given quantity as a single logarithm.

1/5 ln(x + 2)^5 + 1/2 [ln x − ln(x^2 + 3x + 2)^2]

Answer 1

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

Answer 2

All of them are base e, surely?

Answer 3

oops my bad

Answer 4

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

Answer 5

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

Answer 6

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

Answer 7

do you know the property of logarithm ?

so like ln a + ln b = ?
or ln a -ln b = ?
or ln a^2 = ?

Answer 8

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

Answer 9

i'm confused

Answer 10

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

yes ?

Answer 11

yes

Answer 12

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

Answer 13

or ln a - ln b =

Answer 14

ln(a/b)

Answer 15

yes and ln a^2 =

Answer 16

2ln a

Answer 17

yes right sure

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

Answer 18

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

Answer 19

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

Answer 20

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 ?

Answer 21

yess

Answer 22

so than how will be ?

Answer 23

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

Answer 24

why ? than you know that (a^2)^(1/2) =a^(2/2) =a

so than what will be ?

Answer 25

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

Answer 26

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

Answer 27

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 ?

Answer 28

do you understand it sure ?

Answer 29

no

Answer 30

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 ?

Answer 31

so than will result there ln (x+2) + ln (x^1/2) /(x^2 +3x +2)

so from this can you continue it ?

Answer 32

are you here ?

Answer 33

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)

Answer 34

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

Answer 35

do you know ?

Answer 36

so x^2 +3x +2 = (x+2)(x+1)

is right ?

Answer 37

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 ?