Positive logarithmic moments of reciprocal primes (${\sum }_{p\le x}(\log p{)}^{\lambda }/p=(\log x{)}^{\lambda }/\lambda +O((\log x{)}^{\lambda -1})$)

For fixed $\lambda >1$, integrate $(\log t{)}^{\lambda -1}$ against $A(t)={\sum }_{p\le t}(\log p)/p$. The Mertens first theorem gives $A(t)=\log t+O(1)$. Partial summation makes the main integral $(\log x{)}^{\lambda }/\lambda$; the bounded error in $A$ contributes at most a constant times $(\log x{)}^{\lambda -1}$. The restriction $\lambda >1$ ensures that integrating the derivative $(\lambda -1)(\log t{)}^{\lambda -2}/t$ has this bound.