Express the given quantity as a single logarithm.
1/3ln(x + 2)3 + 1/2[ln(x)− ln(x^2+3x+2)^2]
ln a + ln b+ klnc - ln d
ln [abck / d]
A bit of manipulation reveals that it's just ln(sqrt(x)/(x+1))
To express the given quantity as a single logarithm, we can use the logarithm properties.
First, let's simplify the expression step by step.
1/3ln(x + 2)^3 + 1/2[ln(x) − ln(x^2 + 3x + 2)^2]
Using the power property of logarithms, we can rewrite (x + 2)^3 as ln[(x + 2)^3], and (x^2 + 3x + 2)^2 as ln[(x^2 + 3x + 2)^2]:
1/3 ln[(x + 2)^3] + 1/2[ln(x) − ln[(x^2 + 3x + 2)^2]]
Now, we can apply the product property of logarithms to the second term inside the square brackets, which is ln(x) - ln[(x^2 + 3x + 2)^2]:
ln[(x + 2)^3] + 1/2[ln(x) - ln(x^2 + 3x + 2)^2]
Using the quotient property of logarithms, we can combine ln(x) and -ln[(x^2 + 3x + 2)^2]:
ln[(x + 2)^3] + 1/2ln(x / (x^2 + 3x + 2)^2)
Now, we can simplify the expression further. Let's find a common denominator for 1/2 and 2:
ln[(x + 2)^3] + ln[(x / (x^2 + 3x + 2)^2)^(1/2)]
Using the power property of logarithms, we can simplify (x / (x^2 + 3x + 2)^2)^(1/2) as ln[(x / (x^2 + 3x + 2)^2)^(1/2)]:
ln[(x + 2)^3] + ln[(x / (x^2 + 3x + 2)^2)^(1/2)]
Now, we apply the sum property of logarithms to combine the two logarithms:
ln[(x + 2)^3(x / (x^2 + 3x + 2)^2)^(1/2)]
Finally, using the power property of logarithms again, we can simplify the expression:
ln[(x + 2)^3(x / (x^2 + 3x + 2)^2)^(1/2)]
So, the given quantity can be expressed as a single logarithm: ln[(x + 2)^3(x / (x^2 + 3x + 2)^2)^(1/2)]