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Unformatted text preview: Notes on the Natural Exponential Function From your experience in high school logarithms are intimately connected to exponentials. This lesson shows how the natural logarithm leads to the idea of the natural exponential. In addition the graph of the natural exponential is developed and the algebra of the natural exponential is derived. The fundamental idea is that the natural exponential function is the inverse function of the natural logarithm function. It will be necessary to remember from high school algebra how a function and its inverse are related. If f denotes a function that has an inverse (denoted by g ), the basic relationship between f and g is: ( 29 y g x = if and only if ( 29 x f y = If a function f is increasing (or decreasing for that matter) on an interval I , then f has a unique inverse function on I . Definition : Consider ( 29 ln f x x = for0 x < < ∞ . Since the natural logarithm function is increasing on the interval ( 29 0, ∞ , it has a unique inverse function called the natural exponential function , denoted by ( 29 ( 29 exp g x x = . Basic Relationship : ( 29 exp y x = if and only if ln x y = for all numbers x and y for which x-∞ < < ∞ and y < < ∞ . Consequences of the Basic Relationship : The domain of the natural exponential function is the interval ( 29 ,-∞ ∞ and the range of the natural exponential function is the interval ( 29 0, ∞ . For x-∞ < < ∞ , ( 29 ( 29 ln exp x x = and for0 x < < ∞ , ( 29 exp ln x x = . Sinceln1 = , ( 29 exp 0 1 = ; sinceln 1 e = , ( 29 exp 1 e = . The graph of the natural exponential function is the reflection about the line y x = of the graph of the natural logarithm. Since the y-axis is an asymptote in the negative direction for the natural logarithm function, the x-axis is an asymptote in the negative direction for the natural exponential function. 1 So far there is no apparent relationship to exponents. This is remedied in the following manner. Let x-∞ < < ∞ and recall that by a Law of logarithmsln ln x e x e x = = . So ( 29 ( 29 ln exp ln x x x e = = . As a consequence of Law 4 of Logarithms (ln ln a b = if an only if a b = ), this equality can hold if and only if ( 29 exp x x e = . Re-interpreting the Basic Relationship : x y e = if and only if ln x y = for all numbers x and y for which x-∞ < < ∞ and y < < ∞ ....
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