Question

How do you expand `log ((x^3 sqrt (x+1))/(x-2)^2)`?

Answer 1

Break down the term into smaller parts, then clean up to get `3logx+(1/2)log(x+1)-2log(x-2)`

Explanation:

Let's start with the original:

`log((x^3sqrt(x+1))/(x-2)^2)`

Right now we have, in essence, 3 log terms - the `x^3` and the square root multiplying each other, and the square term dividing into the numerator. We can break these apart like so:

`logx^3+logsqrt(x+1)-log(x-2)^2`

I'm going to rewrite this to show the square root as an exponent of `1/2`

`logx^3+log(x+1)^(1/2)-log(x-2)^2`

We can now move the exponents to the space in front of the log sign, like this:

`3logx+(1/2)log(x+1)-2log(x-2)`