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

Break down the term into smaller parts, then clean up to get $3 \log x + \left(\frac{1}{2}\right) \log \left(x + 1\right) - 2 \log \left(x - 2\right)$

#### Explanation:

$\log \left(\frac{{x}^{3} \sqrt{x + 1}}{x - 2} ^ 2\right)$

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:

$\log {x}^{3} + \log \sqrt{x + 1} - \log {\left(x - 2\right)}^{2}$

I'm going to rewrite this to show the square root as an exponent of $\frac{1}{2}$

$\log {x}^{3} + \log {\left(x + 1\right)}^{\frac{1}{2}} - \log {\left(x - 2\right)}^{2}$

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

$3 \log x + \left(\frac{1}{2}\right) \log \left(x + 1\right) - 2 \log \left(x - 2\right)$