Properties of Logarithms

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 of1.7(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. Note this is not the same aslog 1.04 – log 3 Use Properties for Solving Exponential Equations • Given • Take log ofboth sides • Use exponent property • Solve for whatwas the exponent

8. 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)

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. 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.