1. Logarithms and Their Properties
Lesson 5.1

2. Recall the Exponential Function
General form
Given the exponent what is the resulting y-value?
Now we look at the inverse of this function
Now we will ask, given the result, what exponent is needed to achieve it?

3. 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

4. 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

5. Properties of Logarithms
Note box on page 154 of text
Most used properties

6. Natural Logarithms We have used base of 10 for logs
Another commonly used base for logs is e
e is an irrational number (as is )
e has other interesting properties
Later to be discovered in calculus
Use ln button on your calculator

7. Properties of the Natural Logarithm
Recall that y = ln x  x = ey
Note that
ln 1 = 0 and ln e = 1
ln (ex) = x (for all x)
e ln x = x (for x > 0)
As with other based logarithms

8. Use Properties for Solving Exponential Equations
Given
Take log of both sides
Use exponent property
Solve for what was the exponent
Note this is not the same as log 1.04 – log 3

9. Misconceptions log (a+b) NOT the same as log a + log b
log (a * b) NOT same as (log a)(log b)
log (a/b) NOT same as (log a)/(log b)
log (1/a) NOT same as 1/(log a)

10. Assignment
Lesson 5.1
Page 185
Exercises 1 – 51 odd