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‚ä‚é‚¢ ƒnƒ€ƒXƒ^[ ƒCƒ‰ƒXƒg ‚©‚í‚¢‚¢-Those who have taken economics courses may remember the equation above, which lists the components of GDP GDP (Y) is the combination of consumption (C), investment (I), government spending (G), and net exports (exports (X) less imports (M))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 ∈ N

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