SOLUTION: Express as a single term  
1/7 ln(x + 2)^7 + 1/2[lnx − ln(x^2 + 3x + 2)^2]
 
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Question 799168: Express as a single term
1/7 ln(x + 2)^7 + 1/2[lnx − ln(x^2 + 3x + 2)^2]
Answer by DrBeeee(684) (Show Source): You can put this solution on YOUR website!
Given;
(1) 1/7ln(x+2)^7 + 1/2[ln(x) - ln(x^2+3x+2)^2]
Using the property
(2) a*ln(b)^c = ln(b)^a*c
and simplifying the second term yields
(3) ln(x+2)^(7/7) + 1/2ln(x) - 1/2ln(x^2+3x+2)^2 or
(4) ln(x+2) + 1/2ln(x) - 1/2ln(x^2+3x+2)^2
Now apply (2) to the second and third terms of (4) to yield
(5) ln(x+2) + ln(sqrt(x)) - ln(x^2+3x+2)^(2/2) or
(6) ln(x+2) + ln(sqrt(x)) - ln(x^2+3x+2)
Now factor the quadratic in the third term to get
(7) ln(x+2) + ln(sqrt(x)) - ln[(x+2)*(x+1)]
Now use the property
(8) ln(a*b) = ln(a) + ln(b)
in the third term of (7) to get
(9) ln(x+2) + ln(sqrt(x)) - ln(x^2) - ln(x+1)
Note that the first and third terms of (9) cancel, leaving
(10) ln(sqrt(x)) - ln(x+1)
Now use the property
(11) ln(a/b) = ln(a) - ln(b)
in (10) to yield the final answer
(12)
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