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 number is an exponent

3. Note: if no base specified, default is base of 10
The Log Function
Try These log39 = ? log232 = ? 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 = 10x y = log x and

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

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

12. 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 10
100
1000
100000
Types of Insects
500
800
1100
1400
1700

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

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

15. Assignment Lesson 5.4A Page 402 Exercises 1 – 71 EOO Lesson 5.4B
Exercises 75 – 89 odd