1. Logarithms and Their Properties Lesson 4.1

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 = 10 x Based on that function, declare a new function x = log 10 y You should be able to see that these are inverse functions In general The log of a number is an exponent

4. The Log Function Try These log 3 9 = ?log 2 32 = ? 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 = 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

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 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 4.1 Page 157 Exercises 1 – 51 odd