equivalent expression for 2log(x+1)-2log(x-1)-log(x-1)
Simplifying
2log(x + 1) + -2log(x + -1) + -1log(x + -1)
Reorder the terms:
2glo(1 + x) + -2log(x + -1) + -1log(x + -1)
(1 * 2glo + x * 2glo) + -2log(x + -1) + -1log(x + -1)
(2glo + 2glox) + -2log(x + -1) + -1log(x + -1)
Reorder the terms:
2glo + 2glox + -2glo(-1 + x) + -1log(x + -1)
2glo + 2glox + (-1 * -2glo + x * -2glo) + -1log(x + -1)
2glo + 2glox + (2glo + -2glox) + -1log(x + -1)
Reorder the terms:
2glo + 2glox + 2glo + -2glox + -1glo(-1 + x)
2glo + 2glox + 2glo + -2glox + (-1 * -1glo + x * -1glo)
2glo + 2glox + 2glo + -2glox + (1glo + -1glox)
Reorder the terms:
2glo + 2glo + 1glo + 2glox + -2glox + -1glox
Combine like terms: 2glo + 2glo = 4glo
4glo + 1glo + 2glox + -2glox + -1glox
Combine like terms: 4glo + 1glo = 5glo
5glo + 2glox + -2glox + -1glox
Combine like terms: 2glox + -2glox = 0
5glo + 0 + -1glox
5glo + -1glox
log [ (x+1)^2 /((x-1)^2(x-1)) ]
= log ((x+1)^2 / (x-1)^3 )
To find the equivalent expression for 2log(x+1) - 2log(x-1) - log(x-1), we can use logarithmic properties.
First, let's rewrite the given expression using the properties of logarithms:
2log(x+1) - 2log(x-1) - log(x-1)
Using the power rule of logarithms, we can rewrite 2log(x+1) as log((x+1)^2):
log((x+1)^2) - 2log(x-1) - log(x-1)
Next, using the quotient rule of logarithms, we can rewrite 2log(x-1) as log((x-1)^2):
log((x+1)^2) - log((x-1)^2) - log(x-1)
Now, using the rule of logarithms that states log(a) - log(b) is equal to log(a/b), we can combine the terms as follows:
log((x+1)^2 / (x-1)^2) - log(x-1)
Finally, using the rule of logarithms that states log(a) - log(b) = log(a/b), we can further simplify the expression:
log((x+1)^2 / (x-1)^2(x-1))
Therefore, the equivalent expression for 2log(x+1) - 2log(x-1) - log(x-1) is:
log((x+1)^2 / (x-1)^2(x-1))