equivalent expression for 2log(x+1) - log(x^2 -1) +log(x-1)
since x^2-1 = (x-1)(x+1)
log[ (x+1)^2 / ((x-1)(x+1)) * (x-1) ]
= log(x+1)
To find the equivalent expression for 2log(x+1) - log(x^2 -1) + log(x-1), we can use logarithmic properties to simplify the expression.
First, let's go through each term one by one:
1) 2log(x+1)
Using the logarithmic property log(a^b) = b*log(a), we can rewrite the term as log((x+1)^2).
2) -log(x^2 -1)
Using the logarithmic property log(a/b) = log(a) - log(b), we can rewrite the term as log(1/(x^2 - 1)).
3) log(x-1)
Now, we can put all the terms together:
Equivalent expression = log((x+1)^2) + log(1/(x^2 - 1)) + log(x-1).
Next, we can simplify the expression further by using logarithmic properties:
1) Using the logarithmic property log(a) + log(b) = log(a*b), we can combine the first two terms:
Equivalent expression = log(((x+1)^2) * (1/(x^2 - 1))) + log(x-1).
2) Using the logarithmic property log(a) + log(b) = log(a*b), we can further simplify the expression:
Equivalent expression = log((((x+1)^2) * (1/(x^2 - 1))) * (x-1)).
3) Using the logarithmic property log(a * b * c) = log(a) + log(b) + log(c), we can further simplify the expression:
Equivalent expression = log(((x+1)^2) * (1/(x^2 - 1)) * (x-1)).
So, the equivalent expression for 2log(x+1) - log(x^2 -1) + log(x-1) is log(((x+1)^2) * (1/(x^2 - 1)) * (x-1)).