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) $log_{a^k}$($x^m$) = $\frac{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.