Understanding Common and Natural Logarithmic Functions

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