1 / 13

§ 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

michi
Download Presentation

§ 4.4

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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).

More Related