1. Properties of Logarithms Pre-Calculus Teacher – Mrs.Volynskaya

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

3. 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

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

5. 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

6. Misconceptions log (a+b) NOT the same as log a + log b 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)

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

8. 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

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

10. 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

11. Decibels Suppose I 0 is the softest sound the human ear can hear measured in watts/cm 2 And I is the watts/cm 2 of a given sound Then the decibels of the sound is The log of the ratio

12. 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

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

14. 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

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