Question

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

Answer 1

`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)`