Exponentials

For all real a > 0, m and n,

a0 = 1 a1 = a -1 1- a = a (am )n = amn am ⋅an = am+n (1.2.2)

For all n and a ≥ 1, the function aⁿ is monotonically increasing in n. For convenience, we assume 0⁰ = 1.

Natural Logarithm Function

e = 2.71828…, the base of the natural logarithm function.

x ∞∑ xi e = (-i!) i=0 = lim (1 + x)n (1.2.3) n→ ∞ n

For all real x we have the inequality

x x 2 e ≥ 1+ x,x = 01+ x ≤ e ≤ 1+ x + x,|x| ≤ 1

(1.2.4)