Expanding log problem

Question

I found this site with online problems and answers. https://courses.lumenlearning.com/waymakercollegealgebra/chapter/expand-and-condense-logarithms/

I've tried several problems and my answer is always wrong.

I've added two screenshots: 1) the problem and the answer according to the site

2) my work and answer

Could someone please explain what I'm doing wrong? I don't understand why the denominator disappears.

Thanks

Answer 1

Because the domain gives $y>7$ and $$\ln\frac{\sqrt{(y^2-49)(y+8)^6}}{y+7}=\ln\frac{\sqrt{y-7}|y+8|^3}{\sqrt{y+7}}=\frac{1}{2}\ln(y-7)-\frac{1}{2}\ln(y+7)+3\ln|y+8|=$$ $$=\frac{1}{2}\ln(y-7)-\frac{1}{2}\ln(y+7)+3\ln(y+8).$$

Answer 2

Assuming for the definition of the given expression that $y>7$ we can avoid absolute values when extracting from square roots since all factors are positive and your result appears to be correct.