Multivariable asymptotic analysis?

Question

Show that $k \ln k = \Theta (n)$ implies $k = \Theta (n /\ln n)$.

Thanks for the help.

Answer 1

If $k<\delta n/\ln n$ (where $0<\delta<1$) then $\ln k<\ln \delta+\ln n<2\ln n$. Consequently, $$\frac{k\ln k}{n}< \frac{2 k \ln n}{n} <2\delta$$ The latter does not happen for very small $\delta$, though.