Why is binary_crossentropy loss used??

[Ayush-Goyal-coding]

According to the paper, Shouldn't loss of discriminator be log(D(x))+ log(1-D(G(z))
And generator be log(1-D(G(z))
so why the loss in both cases is xlog(x) + (1-x)(log(1-x) i.e binary_crossentropy

If they are equivalent then can you show how??