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§ 4.4

§ 4.4. The Natural Logarithm Function. Section Outline. 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

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§ 4.4

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  1. §4.4 The Natural Logarithm Function

  2. Section Outline • 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 • Max’s and Min’s of Exponential Equations

  3. The Natural Logarithm of x

  4. Properties of the Natural Logarithm

  5. Properties of the Natural Logarithm • The point (1, 0) is on the graph of y = ln x [because (0, 1) is on the graph of y = ex]. • ln x is defined only for positive values of x. • ln x is negative for x between 0 and 1. • ln x is positive for x greater than 1. • ln x is an increasing function and concave down.

  6. Exponential Expressions EXAMPLE Simplify. SOLUTION Using properties of the exponential function, we have

  7. Solving Exponential Equations EXAMPLE Solve the equation for x. SOLUTION This is the given equation. Remove the parentheses. Combine the exponential expressions. Add. Take the logarithm of both sides. Simplify. Finish solving for x.

  8. Solving Logarithmic Equations EXAMPLE Solve the equation for x. SOLUTION This is the given equation. Divide both sides by 5. Rewrite in exponential form. Divide both sides by 2.

  9. Other Exponential and Logarithmic Functions

  10. Common Logarithms

  11. Max’s & Min’s of Exponential Equations EXAMPLE The graph of is shown in the figure below. Find the coordinates of the maximum and minimum points.

  12. Max’s & Min’s of Exponential Equations 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. This is the given function. Differentiate using the product rule. Finish differentiating. Factor. Set the derivative equal to 0. Set each factor equal to 0. Simplify.

  13. Max’s & Min’s of Exponential Equations CONTINUED 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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