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Proof lnexy = xy = lnex lney = ln(ex ·ey) Since lnx is onetoone, then exy = ex ·ey 1 = e0 = ex(−x) = ex ·e−x ⇒ e−x = 1 ex ex−y = ex(−y) = ex ·e−y = ex 1 ey ex ey • For r = m ∈ N, emx = e z }m { x···x = z }m { ex ···ex = (ex)m • For r = 1 n, n ∈ N and n 6= 0, ex = e n n x = e 1 nx n ⇒ e n x = (ex) 1 • For r rational, let r = m n, m, n ∈ NNew Jersey is home to more millionaires per capita than any other stateChapter 8 Euclidean Space and Metric Spaces 81 Structures on Euclidean Space 811 Vector and Metric Spaces The set K n of n tuples x = ( x 1;x 2;xn) can be made into a vector space by introducing the standard operations of addition and scalar multiplication Typo A B C D E F G H I J K L M N O ‰Âˆ¤‚¢ ƒ|[ƒY ƒCƒ‰ƒXƒg ƒAƒCƒRƒ"