1. Properties of Logarithms
Lesson 5.5

2. Basic Properties of Logarithms
Note box on page 408 of text
Most used properties

3. Using the Log Function for Solutions
Consider solving
Previously used algebraic techniques (add to, multiply both sides) not helpful
Consider taking the log of both sides and using properties of logarithms

4. Try It Out Consider solution of 1.7(2.1) 3x = 2(4.5)x Steps
Take log of both sides
Change exponents inside log to coefficients outside
Isolate instances of the variable
Solve for variable

5. 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

6. 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

7. 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

8. 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)

9. Usefulness of Logarithms
Logarithms useful in measuring quantities which vary widely
Acidity (pH) of a solution
Sound (decibels)
Earthquakes (Richter scale)

10. Chemical Acidity pH defined as pH = -log[H+]
where [H+] is hydrogen ion concentration
measured in moles per liter
If seawater is [H+]= 1.1*10-8
then –log(1.1*10-8) = 7.96

11. Chemical Acidity
What would be the hydrogen ion concentration of vinegar with pH = 3?

12. Logarithms and Orders of Magnitude
Consider increase of CDs on campus since 1990
Suppose there were 1000 on campus in 1990
Now there are 100,000 on campus
The log of the ratio is the change in the order of magnitude

13. Decibels Suppose I0 is the softest sound the human ear can hear
measured in watts/cm2
And I is the watts/cm2 of a given sound
Then the decibels of the sound is
The log of the ratio

14. Logarithms and Orders of Magnitude
We use the log function because it “counts” the number of powers of 10
This is necessary because of the vast range of sound intensity that the human ear can hear

15. Decibels
If a sound doubles, how many units does its decibel rating increase?

16. Use base 10 or base e which calculator can do for you
Change of Base Formula
We have used base 10 and base e
What about base of another number
log 2 17 = ?
Use formula
Think how to create a function to do this on your calculator
Use base 10 or base e which calculator can do for you

17. Assignment Lesson 5.5 Page 444 Exercises 1 – 85 EOO
Assign change of base spreadsheet
Due in 1 week.