  We derive a sharp Logarithmic Sobolev inequality with monomial weights
starting from a sharp Sobolev inequality with monomial weights. Several related
inequalities such as Shannon type and Heisenberg's uncertain type are also
derived. A characterization of the equality case for the Logarithmic Sobolev
inequality is given when the exponents of the monomial weights are all zero or
integers. Such a proof is new even in the unweighted case.
