Write the expression below as a single logarithm. Please show all of your work.
ln((x^2 – 1)/(x^2 – 6x + 8)) – ln((x+1)/(x+2))
ln((x^2 – 1)/(x^2 – 6x + 8)) – ln((x+1)/(x+2))
when you subtract the logs, you divide the arguments
ln (x+1)(x-1) / [(x+1)(x-1)((x-4)(x-2)]
ln 1/[(x-4)(x-2)]
ln [ (x-4)(x-2) ]^-1
-ln [ (x-4)(x-2)]
6logb v+2logb t
12logb^2vt
To write the expression as a single logarithm, we can use the properties of logarithms. Specifically, we can use the quotient rule, which states that ln(a/b) is equal to ln(a) minus ln(b).
Let's go step by step to simplify the expression:
1. Start with the given expression:
ln((x^2 – 1)/(x^2 – 6x + 8)) – ln((x+1)/(x+2))
2. Apply the quotient rule of logarithms to the numerator and denominator of the first logarithm:
ln((x^2 – 1)/(x^2 – 6x + 8)) = ln(x^2 – 1) – ln(x^2 – 6x + 8)
3. Apply the quotient rule of logarithms to the numerator and denominator of the second logarithm:
ln((x+1)/(x+2)) = ln(x+1) – ln(x+2)
4. Substitute the above results back into the original expression:
ln(x^2 – 1) – ln(x^2 – 6x + 8) – ln(x+1) + ln(x+2)
5. Combine the logarithms using the subtraction property of logarithms:
ln(x^2 – 1) + ln(x+2) – ln(x^2 – 6x + 8) – ln(x+1)
6. Now, we can combine the logarithms using the addition property of logarithms:
ln((x^2 – 1)(x+2)/(x^2 – 6x + 8)(x+1))
Thus, the expression ln((x^2 – 1)/(x^2 – 6x + 8)) – ln((x+1)/(x+2)) can be simplified as ln((x^2 – 1)(x+2)/(x^2 – 6x + 8)(x+1)).