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Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 1 of 55 § 4.4 The Natural Logarithm Function

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Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 2 of 55 The Natural Logarithm of x Properties of the Natural Logarithm Exponential Expressions Solving Exponential Equations Solving Logarithmic Equations Other Exponential and Logarithmic Functions Common Logarithms Maxs and Mins of Exponential Equations Section Outline

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Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 3 of 55 The Natural Logarithm of x DefinitionExample Natural logarithm of x: Given the graph of y = e x, the reflection of that graph about the line y = x, denoted y = ln x

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Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 4 of 55 Properties of the Natural Logarithm

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Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 5 of 55 Properties of the Natural Logarithm 1)The point (1, 0) is on the graph of y = ln x [because (0, 1) is on the graph of y = e x ]. 2)ln x is defined only for positive values of x. 3)ln x is negative for x between 0 and 1. 4)ln x is positive for x greater than 1. 5)ln x is an increasing function and concave down.

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Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 6 of 55 Exponential ExpressionsEXAMPLE SOLUTION Simplify. Using properties of the exponential function, we have

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Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 7 of 55 Solving Exponential EquationsEXAMPLE SOLUTION Solve the equation for x. This is the given equation. Remove the parentheses. Combine the exponential expressions. Add. Take the logarithm of both sides. Simplify. Finish solving for x.

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Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 8 of 55 Solving Logarithmic EquationsEXAMPLE SOLUTION Solve the equation for x. This is the given equation. Divide both sides by 5. Rewrite in exponential form. Divide both sides by 2.

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Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 9 of 55 Other Exponential and Logarithmic Functions

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Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 10 of 55 Common Logarithms DefinitionExample Common logarithm: Logarithms to the base 10

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Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 11 of 55 Maxs & Mins of Exponential EquationsEXAMPLE The graph of is shown in the figure below. Find the coordinates of the maximum and minimum points.

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Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 12 of 55 Maxs & Mins of Exponential Equations This is the given function. CONTINUED At the maximum and minimum points, the graph will have a slope of zero. Therefore, we must determine for what values of x the first derivative is zero. Differentiate using the product rule. Finish differentiating. Factor. Set the derivative equal to 0. Set each factor equal to 0. Simplify.

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Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 13 of 55 Maxs & Mins of Exponential EquationsCONTINUED Therefore, the slope of the function is 0 when x = 1 or x = -1. By looking at the graph, we can see that the relative maximum will occur when x = -1 and that the relative minimum will occur when x = 1. Now we need only determine the corresponding y-coordinates. Therefore, the relative maximum is at (-1, 0.472) and the relative minimum is at (1, -1).

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