1. Logarithmic Functions
Lesson 2.5

2. How to Graph These Numbers?
Consider the vast range of the numbers
Distance from the Sun
Object
Distance (million km)
Mercury 58
Venus
108
Earth
149
Mars
228
Jupiter
778
Saturn
1426
Uranus
2869
Neptune
4495
Pluto
5900
Proxima Centauri
4.1E+07
Andromeda Galaxy
2.4E+13

3. How to Graph These Numbers?
What's wrong with this picture?

4. How to Graph These Numbers?
What's wrong with this picture?
We need a way to set a scale that fits all the data

5. How to Graph These Numbers?
The solution: Set the scale to be the exponent of the distance
This is called a logarithmic scale

6. 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

7. 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

8. Properties of Logarithms
Note box on page 105 of text
Most used properties

9. Note new spreadsheet assignment on Blackboard
Change of Base Theorem
To find the log of a number for a base other than 10 or e …
Use
Where b can be any base
Typically 10 or e
Available on calculator
Note new spreadsheet assignment on Blackboard

10. Change of Base Theorem Create a function for your calculator
Define function
Try it
Verify

11. Solving Log Equations Use definition of logarithm Result x = 32
Rewrite log equation as an exponential equation
Result x = 32

12. Solving Exponential Equations
Use property of logarithms
Consider
Isolate exponential expression
Take ln of both sides
Solve for x

13. Doubling Time What if inflation is at the 5% rate …
How long until prices double?
Strategy
Divide through by P
Take log of both sides
Bring t out as coefficient
Solve for t

14. Assignment
Lesson 2.5
Page 121
Exercises 1 – 79 EOO