How do you use the properties of logarithms to expand ln((x^2-1)/x^3)?

1 Answer
Nov 19, 2017

ln((x^2-1)/x^3) = ln(x-1) + ln(x+1) - 3ln(x)

Explanation:

If a, b > 0 then:

ln(a/b) = ln(a) - ln(b)

and:

ln(ab) = ln(a)+ln(b)

Hence we also find:

ln(a^n) = ln(overbrace(a * a * ... * a)^"n times") = overbrace(ln(a) + ln(a) + ... + ln(a))^"n terms" = n ln(a)

So:

ln((x^2-1)/x^3) = ln(x^2-1)-ln(x^3)

color(white)(ln((x^2-1)/x^3)) = ln((x-1)(x+1))-3ln(x)

color(white)(ln((x^2-1)/x^3)) = ln(x-1) + ln(x+1) - 3ln(x)