General Logarithmic and Exponential Functions Section 7.4* General Logarithmic and Exponential Functions

GENERAL EXPONENTIAL FUNCTIONS Definition: If a > 0, we define the general exponential function with base a by f (x) = ax = ex ln a for all real numbers x.

NOTES ON f(x) = ax 1. f (x) = ax is positive for all x 2. For any real number r, ln (ar) = r ln a

LAWS OF EXPONENTS If x and y are real numbers and a, b > 0, then

DIFFERENTIATION OF GENERAL EXPONENTIAL FUNCTIONS

ANTIDERIVATIVES OF GENERAL EXPONENTIAL FUNCTIONS

THE GENERAL LOGARITHMIC FUNCTION Definition: If a > 0 and a ≠ 1, we define the logarithmic function with base a, denoted by loga, to be the inverse of f (x) = ax. Thus

NOTES ON THE GENERAL LOGARITHMIC FUNCTION 1. loge x = ln x 2.

THE CHANGE OF BASE FORMULA For any positive number a (a ≠ 1), we have

DIFFERENTIATION OF GENERAL LOGARITHMIC FUNCTIONS

THE GENERALIZED VERSION OF THE POWER RULE Theorem: If n is any real number and f (x) = xn, then

THE NUMBER e AS A LIMIT