General Logarithmic and Exponential Functions

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Section 7.4

• General Logarithmic and Exponential Functions

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GENERAL EXPONENTIAL FUNCTIONS
Definition If a gt 0, we define the general
exponential function with base a by f (x) ax
ex ln a for all real numbers x.
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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
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LAWS OF EXPONENTS
If x and y are real numbers and a, b gt 0, then
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DIFFERENTIATION OF GENERAL EXPONENTIAL FUNCTIONS
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ANTIDERIVATIVES OF GENERAL EXPONENTIAL FUNCTIONS
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THE GENERAL LOGARITHMIC FUNCTION
Definition If a gt 0 and a ? 1, we define the
logarithmic function with base a, denoted by
loga, to be the inverse of f (x) ax. Thus
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NOTES ON THE GENERAL LOGARITHMIC FUNCTION
1. loge x ln x 2.
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THE CHANGE OF BASE FORMULA
For any positive number a (a ? 1), we have
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DIFFERENTIATION OF GENERAL LOGARITHMIC FUNCTIONS
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THE GENERALIZED VERSION OF THE POWER RULE
Theorem If n is any real number and f (x) xn,
then
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THE NUMBER e AS A LIMIT