pk(x)=πk1√2πσkexp(−12σ2k(x−μk)2)∑πl1√2πσlexp(−12σ2l(x−μl)2)log(pk(x))=log(πk)+log(1√2πσk)+−12σ2k(x−μk)2log(∑πl1√2πσlexp(−12σ2l(x−μl)2))log(pk(x))log(∑πl1√2πσlexp(−12σ2l(x−μl)2))=log(πk)+log(1√2πσk)+−12σ2k(x−μk)2δ(x)=log(πk)+log(1√2πσk)+−12σ2k(x−μk)2
As you can see, δ(x) is a quadratic function of x.