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Explore the definitions, laws, differentiation, and antiderivatives of general exponential and logarithmic functions. Understand the generalized power rule and the significance of the number e as a limit. Dive into the change of base formula and the inverse relationships between these functions.
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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) = axis 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
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
THE GENERALIZED VERSION OF THE POWER RULE Theorem: If n is any real number and f (x) = xn, then
