1. Logarithmic Functions and Models Lesson 5.4

2. A New Function Consider the exponential function y = 10 x Based on that function, declare a new function x = log 10 y You should be able to see that these are inverse functions In general The log of a number is an exponent

3. The Log Function Try These log 3 9 = ?log 2 32 = ? log 0.01 = ? Note: if no base specified, default is base of 10

4. Graph, Domain, Range Use your calculator to discover facts about the log function In the Y= screen, specify log(x) Set tables with  T initial x = 0,  x = 0.1 View the tables

5. Graph, Domain, Range Note domain for 0 < x < 1 Change the  x to 5, view again

6. Graph, Domain, Range View graph with window -1 < x < 10, -4 < y < 5 Why does the graph appear undefined for x < 0 ?

7. Graph, Domain, Range Recall that There can be no value for y that gives x < 0 Domain for y = log x x > 0 Range y = { all real values }

8. Vertical Asymptote Note behavior of function as x  0 +

9. Inverse Properties Explain why the following would be true. Note the graphical relationship of y = 10 x y = log x and

10. Assignment Lesson 5.4A Page 433 Exercises 1 – 49 odd

11. Solving Exponential Equations Consider Divide by 2 Take log both sides Rewrite using inverse

12. Try It Out Consider solution of Steps Isolate the 10 x Take log of both sides Use the inverse property

13. Modeling Data with Logarithms Consider the table below We seek to model this data with a function Substitute values to get two equations with a and b – solve the equations Acres 101001000100000 Types of Insects 500800110014001700

14. Modeling Data with Logarithms Substitute values of x and y Now use substitution for a and b Finally f(x) = 200 + 300 log x

15. Logarithmic Equations Consider solving the logarithmic equation log 4x = 2 Exponentiate both sides using the base Use the inverse property … and solve

16. Assignment Lesson 5.4B Page 434 Exercises 53 – 97 odd