1. Mrs. McConaughyHonors Algebra 21 Graphing Logarithmic Functions During this lesson, you will:  Write an equation for the inverse of an exponential or a logarithmic function  Graph a logarithmic function  Read and interpret the graph of a logarithmic function (…maybe!)

2. Mrs. McConaughyHonors Algebra 22 Writing an Equation for the Inverse of an Exponential or a Logarithmic Function Given y = 2 x, write its inverse (logarithmic form) in y = form. y = 2 x x = 2 y ; 2 y = xSwitch x and y. Re-write in logarithmic form. log 2 x = y ; y = log 2 x

3. Mrs. McConaughyHonors Algebra 23 Given log 2 x = y, write its inverse (exponential form) in y = form log 2 x = y Switch x and y. log 2 y = x Re-write in exponential form. 2 x = y; y = 2 x

4. Mrs. McConaughyHonors Algebra 24 Inverse of an Exponential Function Inverse of an Exponential Function By definition, f(x) = b x and g(x) = log b x are inverses of each other.

5. Mrs. McConaughyHonors Algebra 25 Graphs of Exponential and Logarithmic Functions The graph of the logarithmic function is the ________ of the exponential function, which tells us:  the logarithmic function ___________ _____________________________  the graph of the logarithmic function is ______________________________ ______________________________ _____. inverse reverses the coordinates of the exponential function a reflection of the graph of the exponential function about the line y = x.

6. Mrs. McConaughyHonors Algebra 26 Graphing Exponential and Logarithmic Equations Using a Table of Values

7. Mrs. McConaughyHonors Algebra 27 Using a Table of Values x-2012 f(x) = 2 x 1. Determine a table of coordinates for f(x) = 2 x. ¼½124 x g(x) = log 2 x -2 012 ¼½124 2.______________ these coordinates to find coordinates of the inverse function g(x) = log 2 x. Reverse

8. Mrs. McConaughyHonors Algebra 28 3. Graph f(x) = 2 x 4. Graph g(x) = log 2 x. Graphing Exponential and Logarithmic Equations NOTE: The graph of the inverse can also be drawn by ______________________________. reflecting the graph of fx about the line y = x

9. Mrs. McConaughyHonors Algebra 29 Example: Graph y = log 10 x. y = 10 x 2 2

10. Mrs. McConaughyHonors Algebra 210 Summary: Characteristics of the Graphs of Logarithmic Functions  The x-intercept is ___________; there is ________ y-intercept.  The y-axis is a ______________________.  If b > 1, the function is __________________________________. If 0 < b < 1, the function is __________________________________. (1,0) no a vertical asymptote increasing decreasing

11. Mrs. McConaughyHonors Algebra 211 Transformations Involving Logarithmic Functions

12. Mrs. McConaughyHonors Algebra 212 TransformationEquationDescription Horizontal Translation y= log b (x + c)  Shifts log b (x + c) to the left c units if c> 0.  Shifts log b (x + c) to the right c units if c < 0.  Vertical asymptote for both graphs = -c. Vertical Stretching or Shrinking y = clog b x  Multiplying y = log b x by c,  Stretches the graph if c > 1  Shrinks the graph if 0 < c <1 Reflectingy = - log b x y = log b (-x)  Reflects the graph of y = log b x about the x-axis  Reflects the graph of y = log b x about the y-axis Vertical Translation y = c + log b x  Shifts the graph of y = c + log b x upward c units if c >0  Shifts the graph of y = c + log b x downward c units if c<0

13. Mrs. McConaughyHonors Algebra 213 Finding the Domain of a Logarithmic Function The domain of an exponential function includes _____________________ and its range is ____________________________. In general, the domain of of y = log b (x + c) consists of all x for which __________. all real numbers all positive real numbers x+ c > 0

14. Mrs. McConaughyHonors Algebra 214 Homework Assignment: P-H Text, pages 443-444: 63-71 odd; 72-75 all (graph); 75-83 odd (Domain and Range only), 90.