MATH 12002 - CALCULUS I § 5.3: The Natural Exponential Function Professor Donald L. White Department of Mathematical Sciences Kent State University D.L. White (Kent State University) 1 / 7
Definition & Basic Properties Recall f ( x ) = ln x is one-to-one with domain (0 , ∞ ) and range ( −∞ , ∞ ). Therefore, ln x has an inverse, which for now we call will f − 1 ( x ) = exp x , with domain ( −∞ , ∞ ) and range (0 , ∞ ). By the cancellation property of inverses, ln(exp x ) = x for all x . Recall also that we defined the number e with the property that ln e = 1. By the Laws of Logarithms, if x is any real number, then ln( e x ) = x ln e = x · 1 = x . Hence for any real number x , we have ln(exp x ) = ln( e x ). Since ln x is one-to-one, this implies exp x = e x for all x , and so e x is precisely the inverse of our natural logarithm function. In particular, ln( e x ) = x for all x e ln x = x for all x > 0 . The function y = e x is called the natural exponential function. D.L. White (Kent State University) 2 / 7
Definition & Basic Properties As the inverse of y = ln x , the graph of the natural exponential function is the reflection in the line y = x of the graph of y = ln x : § 5.3 FIGURE 1 Thus the graph of y = e x is as in the figure below: § 5.3 FIGURE 2 D.L. White (Kent State University) 3 / 7
Definition & Basic Properties A number of properties of e x follow from properties of its inverse ln x : ⇒ e y = x . ln x = y ⇐ y = e x has domain ( −∞ , ∞ ) and range (0 , ∞ ). That is, e x is defined for all x , but e x is never 0 or negative. e 0 = 1 and e 1 = e , so the points (0 , 1) and (1 , e ) are on the graph. y = ln x is differentiable everywhere and its derivative ( 1 x ) is never 0, hence y = e x is differentiable (so also continuous) everywhere. y = e x is increasing and concave up everywhere. Since y = ln x has vertical asymptote x = 0, y = e x has horizontal asymptote y = 0, i.e., x →−∞ e x = 0. lim x → + ∞ e x = + ∞ . Since x → + ∞ ln x = + ∞ , we also have lim lim D.L. White (Kent State University) 4 / 7
Solving Equations Inverse operations are often used in solving equations: Subtract 5 to solve x + 5 = 3 (inverse of addition). Divide by 5 to solve 5 x = 3 (inverse of multiplication). Take the 5th root to solve x 5 = 3 (inverse of 5th power). We can use e x and ln x to solve logarithmic and exponential equations. Examples: 1 Solve for x : ln(5 x + 2) = 8. Since ln x = y ⇔ e y = x , we can translate this equation to the exponential equation e 8 = 5 x + 2 , e 8 − 2 , 5 x = ( e 8 − 2) / 5 ≈ 595 . 79 . = x D.L. White (Kent State University) 5 / 7
Solving Equations 2 Solve for x : e 2 x − 3 = 4. To get at the variable in the exponent, we apply the inverse function y = ln x to both sides of the equation: ln e 2 x − 3 = ln 4 2 x − 3 = ln 4 = (3 + ln 4) / 2 ≈ 2 . 19 . x 3 Solve for x : 2 x − 3 = 10. Although ln x is not the inverse of 2 x , we can still apply ln x to both sides of the equation and apply Laws of Logarithms: ln 2 x − 3 = ln 10 ( x − 3) ln 2 = ln 10 x − 3 = ln 10 / ln 2 3 + ln 10 x = ln 2 ≈ 6 . 32 . D.L. White (Kent State University) 6 / 7
Evaluating Limits x → + ∞ e x = + ∞ and x →−∞ e x = 0 . We noted previously that lim lim Examples: x → + ∞ e 5 − x 2 . 1 Evaluate the limit lim x → + ∞ e 5 − x 2 = z →−∞ e z = 0. As x → + ∞ , 5 − x 2 → −∞ , so lim lim 2 e x − 5 e − x 2 Evaluate the limit lim 3 e x + 7 e − x . x → + ∞ As x → + ∞ , e x → + ∞ and e − x → 0. Similar to what we did with rational functions, we can divide numerator and denominator by the term that approaches ∞ : (2 e x − 5 e − x ) 1 2 e x − 5 e − x e x lim = lim 3 e x + 7 e − x (3 e x + 7 e − x ) 1 x → + ∞ x → + ∞ e x 2 − 5 e − 2 x 3 + 7 e − 2 x = 2 − 0 3 + 0 = 2 = lim 3 . x → + ∞ D.L. White (Kent State University) 7 / 7
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