Logarithmic Functions and Models

1. Logarithmic Functions and Models Lesson 5.4

2. A New Function • Consider the exponential function y = 10x • Based on that function, declare a new function x = log10y • You should be able to see that these are inverse functions • In general • The log of a numberis an exponent

3. Note: if no base specified, default is base of 10 The Log Function • Try Theselog39 = ? log232 = ? log 0.01 = ?

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 thegraph appearundefinedfor 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 = 10xy = 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 usinginverse

12. Try It Out • Consider solution of • Steps • Isolate the 10x • 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

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