  In this article, we first establish an $L^2$-type Dolbeault isomorphism for
the sheaf of logarithmic differential forms twisted by the multiplier ideal
sheaf. By using this isomorphism and $L^2$-estimates equipped with a singular
Hermitian metric, we obtain logarithmic vanishing theorems involving multiplier
ideal sheaves on compact K\"ahler manifolds with simple normal crossing
divisors.
