Key identities: (1) a^(log_a(x)) = x. (2) a^(log_b(x)) = x^(log_b(a)) (useful for simplification). (3) log_a(b)*log_b(c)*log_c(a) = 1 (cyclic product). (4) log_a(b)*log_b(a) = 1. (5) log_{a^k}(x^m) = (m/k)*log_a(x). (6) If log_a(x) = log_a(y), then x = y (one-to-one property). Identity (2) is derived from taking log_b of both sides: log_b(a^(log_b(x))) = log_b(x)*log_b(a) = log_b(x^(log_b(a))). These identities appear in simplification problems where multiple bases need to be unified.