Express as a single Logarithm and Simplify:
2/3ln(x^2+1)+1/2ln(x^2-1)-1/2ln(x+1)
ln(x^2+1)^(2/3)+ln((X^2-1)^(1/2)/(x+1)^(1/2)) =
ln(x^2+1)^(2/3)+ ln((x+1)(x-1))^(1/2)/
(x+1)^(1/2)) =
ln(x^2+1)^(2/3)+ln(x-1)^(1/2) =
ln((x^2+1)^(2/3)*(x-1)^(1/2)).
OR ln((crt(x^2+1)^2*sqrt(x-1).
To express the given expression as a single logarithm, we can use the properties of logarithms.
First, let's simplify each term separately:
1. Simplifying 2/3ln(x^2+1):
Using the property of logarithms, we can write 2/3ln(x^2+1) as ln((x^2+1)^(2/3)).
2. Simplifying 1/2ln(x^2-1):
Similarly, we can write 1/2ln(x^2-1) as ln((x^2-1)^(1/2)).
3. Simplifying -1/2ln(x+1):
Again, we can write -1/2ln(x+1) as ln((x+1)^(-1/2)).
Now, let's combine these simplified logarithmic expressions into a single logarithm:
ln((x^2+1)^(2/3)) + ln((x^2-1)^(1/2)) - ln((x+1)^(-1/2))
Using the property of logarithms, we can simplify further by combining these logarithms:
ln(((x^2+1)^(2/3)) * ((x^2-1)^(1/2)) / ((x+1)^(-1/2)))
Simplifying the expression inside the logarithm:
ln(((x^2+1)^(2/3)) * ((x^2-1)^(1/2)) * (x+1)^(1/2))
Therefore, the simplified expression, expressed as a single logarithm, is ln(((x^2+1)^(2/3)) * ((x^2-1)^(1/2)) * (x+1)^(1/2)).