Notes on the Generalized Exponential and Logarithmic Functions

Notes on the Generalized Exponential and Logarithmic Functions

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Notes on the Generalized Exponential and Logarithmic Functions It turns out that all exponential and logarithmic functions are related to the Natural Exponential and Natural Logarithm functions. Mathematically, dealing with any exponential is the same as dealing with the Natural Exponential. A similar statement can be made concerning logarithmic functions. This lesson shows why this statement is true. The fundamental idea is that the Natural Logarithm function and the Natural Exponential function are inverse functions for each other. There is little reason to emphasize generalized exponentials and logarithms, but they do arise occasionally. The main things to know are the differentiation and integration formulas as illustrated in the examples. Generalized Exponential Functions Definition : Let 0 a denote a positive number. For any value of the variable x, define ( 29 ln ln x x a x a a e e = = . The number a is called the base of the exponential function . Observations concerning the definition of the generalized exponential function : The exponential function is only defined when the base is a positive number. In other words a function such as ( 29 2 x - is not define. (Actually this is because we deal with real valued functions in this course. If one includes complex variables, then one can discuss such functions.) Since ( 29 ln a x x a e = , all exponential functions are actually powers of the Natural Exponential function. This means that if one wishes to use the function 2 x , then one can use ln2 x e just as well. The domain of all exponential functions is the interval ( 29 , -∞ ∞ . 0 0ln 0 1 a a e e = = = . So for all 0 a the point ( 29 0,1 lies on the graph of the function. If 1 a = , for any value of the variable x , ln1 0 1 1 x x e e = = = . That is to say that the function 1 x is a constant function. So the more interesting cases are when0 1 a < < or 1 a . Properties of Generalized Exponential Functions : Listed below are the primary properties of generalized exponential functions. The verification of these properties is given at the end of these notes. It is not necessary that you memorize how to verify the properties, but it is a good idea to see why they hold. 1. Rules of Exponents : Here 0 a and x and y are any real numbers. Notice that these Rules are the same as those for the Natural Exponential. i. x y x y a a a + = ii. x x y y a a a - = iii. ( 29 y x xy a a = 1
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2. Differentiation and Integration of Generalized Exponential Functions : i. ln x x d a a a dx = and by the Chain Rule ( 29 ( 29 ( 29 ( 29 ln g x g x d d a a a g x dx dx = ii. ln x x a a dx C a = + for some constant C provided 1 a . 3. The Asymptotic Behavior of Generalized Exponential Functions : i. For 0 a , , 1 lim 1, 1 0, 1 x x a a a a →-∞ < = = ii. For 0 a , 0, 1 lim 1, 1 , 1 x x a a a a →∞ < = = 4. Graphs for Generalized Exponential Functions : Notice that there is an interesting issue that arises from the calculus of generalized
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Notes on the Generalized Exponential and Logarithmic Functions

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