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6. The Logarithm, the Exponential, and the Inverse Trigonometric Functions.

Definition. If x is a positive real number, we define the natural logarithm of x, denoted temporarily by L(x), to be the integral L(x) = .

Theorem 6.1. The logarithm function has the following properties:
(a) L(1) = 0
(b) L`(x) = 1/x for every x > 0
(c) L(ab) = L(a) + L(b) for every a > 0, b > 0
- t/a

Theorem. 6.2. For every real number b there is exactly one positive real number a whose logarithm, L(a) is equal to b.

Definition. We denote by e that number for which L(e) = 1.

Definition. If b > 0, b ≠ 1, and if x > 0, the logarithm of x to the base b is the number
logbx = . where the logarithms on the right are natural logarithms.

Point. ∫ 1/x dx = log x + C
Point. ∫ du / u = log u +C
Point. ∫ f`(x)dx / f(x) = log f(x) + C.

Point. L0(x) = log|x| = .
Definition. For any real x, we define E(x) to be that number y whose logarithm is x. That is, y=E(x) means that L(…