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Properties

Powers obey the usual algebraic identity b = b b. In other words, the function

defined by f (k) = b is a group homomorphism from the integers Z under addition onto the subgroup H of G generated by b. For all a in H, log a exists. Conversely, log a does not exist for a that are not in H.

If H is infinite, then log a is also unique, and the discrete logarithm amounts to a group isomorphism

On the other hand, if H is finite of order n, then log a is unique only up to congruence modulo n, and the discrete logarithm amounts to a group isomorphism

where Z denotes the additive group of integers modulo n.

The familiar base change formula for ordinary logarithms remains valid: If c is another generator of H, then