Multiplicative Functions of Numbers Set and Logarithmic Identities. Shannon and factorial logarithmic Identities, Entropy and Coentropy

The multiplicative functions characterizing the finite set of positive numbers are introduced in the work. With their help we find the logarithmic identities which connect logarithm of sum of the set numbers and logarithms of numbers themselves. One of them (contained in the work of Shannon) interconnects three information functions: information Hartley, entropy and coentropy. Shannon's identity allows better to understand the meaning and relationship of these collective characteristics of information (as the characteristics of finite sets and as probabilistic characteristics). The factorial multiplicative function and the logarithmic factorial identity are formed also from initial set numbers. That identity connects logarithms of factorials of integer numbers and logarithm of factorial of their sum.