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Section 7.4*

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 ) = a x = e x ln a for all real numbers x. NOTES ON f ( x ) = a x.

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Section 7.4*

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  1. Section 7.4* General Logarithmic and Exponential Functions

  2. 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.

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

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

  5. DIFFERENTIATION OF GENERAL EXPONENTIAL FUNCTIONS

  6. ANTIDERIVATIVES OF GENERAL EXPONENTIAL FUNCTIONS

  7. 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

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

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

  10. DIFFERENTIATION OF GENERAL LOGARITHMIC FUNCTIONS

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

  12. THE NUMBER e AS A LIMIT

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