Summation notation conversion to multiplication in attached equation.. How? Please explain.

Question

Can any one please tell me how summation sign has been changed to multiplication in 2nd and 3rd equation and inequality also changed to equal sign.

Answer 1

This is just using the fact that $\log(a\cdot b) = \log a + \log b$ (true for any particular base of the logarithm). Stated more generally, it becomes $\log \prod_{\alpha}x_{\alpha} = \sum_{\alpha}\log x_{\alpha}$.

That particular law of logarithms is a restatement of the fact that $x^u\cdot x^v = x^{u+v}$.