Q:

Express the quantity as a single logarithm: $ \frac 13 \ln (x+2)^3 + \frac 12[ \ln x - \ln (x^2+3x+ 2)^2]$

Express the quantity as a single logarithm: $ \frac 13 \ln (x+2)^3 + \frac 12[ \ln x - \ln (x^2+3x+ 2)^2]$

The answer in the book is ln $\frac {\sqrt{x}}{x+1}$
If am not allowed to to cancel terms when they are sums, how was this answered calculated? 

A:

for $x>0$ we get $$\log(x+2)+\log(x^{1/2})-\log(x^2+3x+2)=\log((x+2)x^{1/2})-\log(x^2+3x+2)$$ and finally we get $$\log\left(\frac{(x+2)x^{1/2}}{x^2+3x+2}\right)$$

