1 / 19

5.4 Common and Natural Logarithmic Functions

5.4 Common and Natural Logarithmic Functions. Do Now Solve for x. 1. 5 x =25 2. 4 x =2 3. 3 x =27 4. 10 x =130. 5.4 Common and Natural Logarithmic Functions. Do Now Solve for x. 1. 5 x =25 x=2 2. 4 x =2 x= ½ 3. 3 x =27 x=3 4. 10 x =130 x≈2.11. Common Logarithms.

kezia
Download Presentation

5.4 Common and Natural Logarithmic Functions

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. 5.4 Common and Natural Logarithmic Functions Do Now Solve for x. 1. 5x=25 2. 4x=2 3. 3x=27 4. 10x=130

  2. 5.4 Common and Natural Logarithmic Functions Do Now Solve for x. 1. 5x=25 x=2 2. 4x=2 x= ½ 3. 3x=27 x=3 4. 10x=130 x≈2.11

  3. Common Logarithms • The inverse function of the exponential function f(x)=10x is called the common logarithmic function. • Notice that the base is 10 – this is specific to the “common” log • The value of the logarithmic function at the number x is denoted as f(x)=log x. • The functions f(x)=10x and g(x)=log x are inverse functions. • log v = u if and only if 10u = v • Notice that the base is “understood “to be 10. • Because exponentials and logarithms are inverses of one another, what do we know about their graphs?

  4. Common Logarithms • Since logs are a special kind of exponent, each logarithmic statement can be expressed as an exponential.

  5. Example 1: Evaluating Common Logs • Without using a calculator, find each value. • log 1000 • log 1 • log 10 • log (-3)

  6. Example 1: Solutions • Without using a calculator, find each value • log 1000  10x = 1000  log 1000 = 3 • log 1  10x = 1  log 1 = 0 • log 10  10x = 10  log 10 = 1/2 • log (-3)  10x = -3  undefined

  7. Evaluating Logarithms • A calculator is necessary to evaluate most logs, but you can get a rough estimate mentally. • For example, because log 795 is greater than log 100 = 2 and less than log 1000 = 3, you can estimate that log 795 is between 2 and 3, and closer to 3.

  8. Using Equivalent Statements • A method for solving logarithmic or exponential equations is to use equivalent exponential or logarithmic statements. • For example: • To solve for x in log x = 2, we can use 102 = x and see that x = 100 • To solve for x in 10x = 29, we can use log 29 = x, and using a calculator to evaluate shows that x = 1.4624

  9. Example 2: Using Equivalent Statements • Solve each equation by using an equivalent statement. • log x = 5 • 10x = 52

  10. Example 2: Solution • Solve each equation by using an equivalent statement. • log x = 5 105 = x x = 100,000 • 10x = 52 log 52 = x x ≈ 1.7160

  11. Natural Logarithms • The exponential function f(x)=ex is useful in science and engineering. Consequently, another type of logarithm exists, where the base is e instead of 10. • The inverse function of the exponential function f(x)=ex is called the natural logarithmic function. • The value of this function at the number x is denoted as f(x)=ln x and is called the natural logarithm.

  12. Natural Logarithms • The functions f(x)=ex and g(x)=ln x are inverse functions. • ln v = u if and only if eu = v • Notice that the base is “understood” to be e. • Again, as with common logs, every natural logarithmic statement is equivalent to an exponential statement.

  13. Example 3: Evaluating Natural Logs • Use a calculator to find each value • ln 1.3 • ln 203 • ln (-12)

  14. Example 3: Solutions • Use a calculator to find each value • ln 1.3 .2624 • ln 203 5.3132 • ln (-12) undefined Why is this undefined??

  15. Example 4: Solving by Using and Equivalent Statement • Solve each equation by using an equivalent statement. • ln x = 2 • ex = 8

  16. Example 4: Solutions • Solve each equation by using an equivalent statement. • ln x = 2 e2 = x  x = 7.3891 • ex = 8 ln8 = x  x = 2.0794

  17. Graphs of Logarithmic Functions • The following table compares graphs of exponential and logarithmic functions (page 359 in your text):

  18. Example 5: Transforming Logarithmic Functions • Describe the transformation of the graph for each logarithmic function. Identify the domain and range. • 3log(x+4) • ln(2-x)-3

  19. Example 5: Transforming Logarithmic Functions • Describe the transformation of the graph for each logarithmic function. Identify the domain and range. • 3log(x+4) Shifted to the left 4 units; vertically stretched by 3 Domain: x > -4 Range: All real numbers • ln(2-x)-3 = ln(-(x-2))-3 Horizontal reflection across y-axis; 2 units to the right; 3 units down Domain: x > 2 Range: All real numbers

More Related