We give alternate proofs for three related results in analysis of Boolean functions, namely the KKL
Theorem, Friedgut’s Junta Theorem, and Talagrand’s strengthening of the KKL Theorem. We follow a
new approach: looking at the first Fourier level of the function after a suitable random restriction and
applying the Log-Sobolev inequality appropriately. In particular, we avoid using the hypercontractive inequality
that is common to the original proofs. Our proofs might serve as an alternate, uniform exposition
to these theorems and the techniques might benefit further research.