For x∈N, x>1 , and p=logxx+1 , q=logx+1x+2 then which one of the following is correct?

$\frac{k}{l}>\frac{k+1}{l+1}$ for (k,l) > 0 and  k > l 

Let     k = x+1    and   l = x

Therefore,  $\frac{x+1}{x}>\frac{(x+1)+1}{\left(x\right)+1}$

 (x + 1) > x

Therefore,  $\frac{\log (x+1)}{\log \left(x\right)}>\frac{\log (x+2)}{\log (x+1)}$

$\Rightarrow \log xx+1 >\log x+1x+2$