log(x^2+2x-3/x^2-4)-log(x^2+7x+6/x+2)
To simplify the expression log(x^2+2x-3/x^2-4) - log(x^2+7x+6/x+2), we can use logarithmic properties.
First, let's focus on the division in each logarithm separately:
log(x^2+2x-3/x^2-4) = log((x^2+2x-3) / (x^2-4))
log(x^2+7x+6/x+2) = log((x^2+7x+6) / (x+2))
Now, recall that the difference of logarithms is equal to the logarithm of the quotient:
log(a) - log(b) = log(a/b)
Applying this property, we can rewrite the expression as:
log((x^2+2x-3) / (x^2-4)) - log((x^2+7x+6) / (x+2))
Now, we can combine the two logarithms into a single logarithm using the division property:
log(((x^2+2x-3) / (x^2-4)) / ((x^2+7x+6) / (x+2)))
To simplify this further, we can simplify the numerator and denominator:
Numerator:
(x^2+2x-3) / (x^2-4) can be factored as ((x+3)(x-1)) / ((x-2)(x+2))
Denominator:
(x^2+7x+6) / (x+2) can be factored as ((x+1)(x+6)) / (x+2)
Substituting the factored forms back into the expression, we have:
log((((x+3)(x-1)) / ((x-2)(x+2))) / (((x+1)(x+6)) / (x+2)))
Next, we can simplify further by cancelling out common factors in the numerator and denominator:
log(((x+3)(x-1)(x+2)) / ((x-2)(x+1)(x+6)))
And that is the simplified form of the expression log(x^2+2x-3/x^2-4) - log(x^2+7x+6/x+2).