1. M3U7D3 Warm Up x = 2 Solve each equation. 1. 8 = x 3 2. x ½ =4 3. 27 = 3 x 4. 4 6 = 4 3x Graph the following: 5. y = 2x 2 x = 16 x = 3 x = 2

2. Homework Check: Document Camera

3. U7D3 Log Properties OBJ: For exponential models, express as a logarithm the solution to ab ct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology. F-LE.4

4. First let’s summarize the properties we discovered observing the classwork we completed and checked in lesson 7

5. NOTICE!!! 2 0 = 1Log 2 1 = 0 2 1 = 2Log 2 2 = 1 2 2 = 4Log 2 4 = 2 2 3 = 8Log 2 8 = 3 2 4 = 16Log 2 16 = 4 2 5 = 32Log 2 32 = 5

6. Properties of Logarithms There are four basic properties of logarithms that we will be working with. For every case, the base of the logarithm can not be equal to 1 and the values must all be positive (no negatives in logs)

7. Product Rule log b MN = log b M + log b N Ex: log b xy = log b x + log b y Ex: log6 = log 2 + log 3 Ex: log 3 9b = log 3 9 + log 3 b

8. Quotient Rule

9. Power Rule

10. Change of Base Formula log 5 8 =Example log 5 8 = This is also how you graph in another base. Enter y 1 =log(8)/log(5). Remember, you don’t have to enter the base when you’re in base 10!

11. These next two problems tend to be some of the trickiest to evaluate. Actually, they are merely identities and the use of our simple rule will show this.

12. Example 1: Solution: Now take it out of the logarithmic form and write it in exponential form. First, we write the problem with a variable.

13. Example 2: Solution: First, we write the problem with a variable. Now take it out of the exponential form and write it in logarithmic form.

14. Ask your teacher about the last two examples. They may show you a nice shortcut. If Log a a b = y then y = b AND… If a Log a b = y then y = b

15. Finally, we want to take a look at the Property of Equality for Logarithmic Functions. Basically, with logarithmic functions, if the bases match on both sides of the equal sign, then simply set the arguments equal. Basically, with logarithmic functions, if the bases match on both sides of the equal sign, then simply set the arguments equal.

16. Example 3: Solution: Since the bases are both ‘3’ we simply set the arguments equal.

17. Example 4: Solution: Since the bases are both ‘8’ we simply set the arguments equal. Factor Solution: But we’re not finished…

18. Example 4 continued… It appears that we have 2 solutions here. If we take a closer look at the definition of a logarithm however, we will see that not only must we use positive bases, but also we see that the arguments must be positive as well. Therefore -2 is not a solution. Let’s end this lesson by taking a closer look at this.

19. Our final concern then is to determine why logarithms like the one below are undefined. Can anyone give us an explanation ?

20. One easy explanation is to simply rewrite this logarithm in exponential form. We’ll then see why a negative value is not permitted. First, we write the problem with a variable. Now take it out of the logarithmic form and write it in exponential form. What power of 2 would gives us -8 ? Hence expressions of this type are undefined. Example 5:

21. Natural Logs A logarithm to the base e (2.71828…). Written ln (pronounced ell-n) Can be accessed on your calculator using LN or 2 nd LN to get to e x.

22. Natural Logs and “e” Start by graphing y=e x The function y=e x has an inverse called the Natural Logarithmic Function. Y=ln x

23. What do you notice about the graphs of y=e x and y=ln x? y=e x and y=ln x are inverses of each other! We can use the natural log to “undo” the function y= e x (and vice versa).

24. All the rules still apply You can use your product, power and quotient rules for natural logs just like you do for regular logs Example 6:

25. Example 7: OR

26. That concludes our introduction to logarithms. In the lessons to follow we will learn some important properties of logarithms. One of these properties will give us a very important tool which we need to solve exponential equations. Until then let’s practice with the basic themes of this lesson. That concludes our introduction to logarithms. In the lessons to follow we will learn some important properties of logarithms. One of these properties will give us a very important tool which we need to solve exponential equations. Until then let’s practice with the basic themes of this lesson.

27. Sum of Properties of General and Natural Logarithms General PropertiesNatural Logarithms 1. log b 1 = 01. ln 1 = 0 2. log b b = 12. ln e = 1 3. log b b x = 03. ln e x = x 4. b log b x = x 4. e ln x = x REMEMBER Common Logarithms are logs base 10.

28. Study Island Study Island is set up and assignments are inserted. You may complete these for points back on low HW, CW, or quizzes. Your login is SID@athens and password is SID. CHECK THIS!!! I think I activated it properly this time.

29. Classwork M3U7D3 Investigating the Properties of Logarithms part I Homework M3U7D3 packet pages 3&4 Properties of Logarithms ALL