1. Definition of a Logarithmic Function For x > 0 and b > 0, b≠ 1, y = log b x is equivalent to b y = x The function f (x) = log b x is the logarithmic function with base b. Date: 3.3: Logarithmic Functions and Their Graphs (3.3) Logarithmic: y = log b x Exponential: b y = x. Form Logarithmic: y = log b x Exponential: b y = x. Form Exponent Base

2. Text Example Write each equation in its equivalent exponential form. a. 3 = log 7 x b. 2 = log b 25c. log 4 26 = y y = log b x means b y = x or y = log 4 26 7 3 = xb 2 = 25 4 y = 26 Write each equation in its equivalent logarithmic form. a. 12 2 = x b. b 3 = 8 c. e y = 9 2 = log 12 x 3 = log b 8 y = log e 9

3. Evaluate a. log 2 16b. log 3 9 c. log 25 5 Solution log 25 5 = 1/2 because 25 1/2 = 5. 25 to what power is 5? log 3 9 = 2 because 3 2 = 9. 3 to what power is 9? log 2 16 = 4 because 2 4 = 16.2 to what power is 16? c. log 25 5 b. log 3 9 a. log 2 16 Logarithmic Expression Evaluated Question Needed for Evaluation Logarithmic Expression Text Example = 4= 2 = 1/2

4. Basic Logarithmic Properties Involving One log b b = 1 because 1 is the exponent to which b must be raised to obtain b (b 1 = b) log b 1 = 0 because 0 is the exponent to which b must be raised to obtain 1 (b 0 = 1)

5. Inverse Properties of Logarithms For x > 0 and b  1, log b b y = y The logarithm with base b of b raised to a power equals that power. b log b x = x b raised to the logarithm with base b of a number equals that number.

6. Characteristics of the Graphs of Logarithmic Functions of the Form f(x) = log b x The x-intercept is 1. There is no y-intercept. The y-axis is a vertical asymptote. If b > 1, the function is increasing. If 0 < b < 1, the function is decreasing. The graph is smooth and continuous. It has no sharp corners or edges. -2 6 2345 5 4 3 2 -2 6 f (x) = 2 x g (x) = log 2 x y = x

7. -2 6 2345 5 4 3 2 -2 6 f (x) = 2 x g (x) = log 2 x y = x Graph f (x) = 2 x and g(x) = log 2 x in the same rectangular coordinate system. SolutionWe first set up a table of coordinates for f (x) = 2 x. Reversing these coordinates gives the coordinates for the inverse function, g(x) = log 2 x. 4 2 8211/21/4f (x) = 2 x 310-2x 2 4 310-2g(x) = log 2 x 8211/21/4x Reverse coordinates. Text Example Graph f (x) = 2 x and g(x) = log 2 x in the same rectangular coordinate system. The graph of the inverse can also be drawn by reflecting the graph of f(x) = 2 x over the line y = x.

8. Complete Student Checkpoint Write each equation in its equivalent logarithmic form. a. 2 5 = x b. b 3 = 27 c. e y = 33 5 = log 2 x 3 = log b 27 y = log e 33 Evaluate a. log 10 100b. log 3 3 c. log 36 6 = 2= 1 = 1/2 10 ? = 100 3 ? = 3 36 ? = 6

9. Complete Student Checkpoint Graph and on the same graph x y -21/9 1/3 01 13 29 f(x) For h(x) just interchange the x and y values of f(x) f(x) -21/9 1/3 01 13 29 x y h(x)

10. What is the Domain for the following functions? (remember you can’t take the log of a negative number) Finding Domains f(x) = log 2 x g(x) = log 2 (x-1) h(x) = log 2 (-x) j(x) = - log 2 x p(x) = log 2 x + 1 r(x) = 2log 2 x x > 0 x > 1 x < 0 x > 0 g(x) = log 4 (x+3) g(x) = log 7 (x-2) g(x) = ln(3-x) g(x) = ln(x-2) 2 x > -3 x > 2 x < 3 all real numbers Complete Student Checkpoint Find the domain of x - 5 >0 x > 5

11. Transformations Involving Logarithmic Functions Shifts the graph of f (x) = log b x upward c units if c > 0. Shifts the graph of f (x) = log b x downward c units if c < 0. g(x) = log b x+c Vertical translation Reflects the graph of f (x) = log b x about the x-axis. Reflects the graph of f (x) = log b x about the y-axis. g(x) = - log b x g(x) = log b ( - x) Reflecting Multiply y-coordintates of f (x) = log b x by c, Stretches the graph of f (x) = log b x if c > 1. Shrinks the graph of f (x) = log b x if 0 < c < 1. g(x) = c log b x Vertical stretching or shrinking Shifts the graph of f (x) = log b x to the left c units if c > 0, asymptote x = -c Shifts the graph of f (x) = log b x to the right c units if c < 0, asymptote x = c g(x) = log b (x+c) Horizontal translation DescriptionEquation Transformation

12. Graph f (x) = log 2 x and g(x) = 1 + -2log 2 (x - 1) in the same rectangular coordinate system. Text Example 2 1 0 4 2 1 1/2 y x f(x)f(x) f(x)f(x) g(x)g(x) What are the transformations for g(x)? Right 1 Stretch x2 Reflect over x-axis Up 1

13. Properties of Common Logarithms (base is 10, log 10 ) General PropertiesCommon Logarithms 1. log b 1 = 01. log 1 = 0 2. log b b = 12. log 10 = 1 3. log b b x = x3. log 10 x = x 4. b log b x = x 4. 10 log x = x The Log on your calculator is base 10

14. Examples of Logarithmic Properties log b b = 1 log b 1 =0 log 4 4 =1 log 8 1 =0 3 log 3 6 =6 log 5 5 3 =3 2 log 2 7 =7

15. Properties of Natural Logarithms ln = log e General PropertiesNatural Logarithms 1. log b 1 = 01. ln 1 = 0 2. log b b = 12. ln e = 1 3. log b b x = x3. ln e x = x 4. b log b x = x 4. e ln x = x

16. Examples of Natural Logarithmic Properties ln e = ln 1 =0 e ln 6 =6 ln e 3 =3 log e e 1 log e 1 e log e 6 log e e 3 Complete Student Checkpoint Use inverse properties to solve:

17. Logarithmic Functions and Their Graphs