Properties of Logarithms

1. Properties of Logarithms Tools for solving logarithmic and exponential equations

2. Let’s review some terms.When we write log5 1255 is called the base125 is called the argument

3. Logarithmic form of 52= 25 islog525 = 2

4. For all the lawsa, M and N > 0 a≠ 1 r is any real

5. Remember ln and log • ln is a short cut for loge • log means log10

6. Easy ones first : loga1 = 0 since a0 = 1

7. log 31= ?

8. log 31= ? loga1 = 0

9. log 31= 0 loga1 = 0

10. ln 1 = ?

11. ln 1= ? loga1 = 0

12. ln 1= 0 loga1 = 0

13. Another easy one : logaa = 1 since a1 = a

14. log 55 = ?

15. log 55 = ? logaa = 1

16. log 55= 1 logaa = 1

17. ln e= ?

18. ln e= logee = ? ln means loge

19. ln e= logee = ? logaa = 1

20. ln e= 1 logaa = 1

21. Just a tiny bit harder : logaar = r since ar = ar

22. ln e3x = ?

23. ln e3x = loge e3x= ? ln means loge

24. ln e3x = loge e3x= ?

25. ln e3x = loge e3x= 3x

26. log(105y)= ?

27. log(105y)= ? log means log10

28. log(105y)=log10 105y= ? log means log10

29. log(105y)=log10 105y= ?

30. log(105y)=log10 105y= ?

31. log(105y)=log10 105y= 5y

32. Evidence that it works (not a proof):

33. Evidence that it works (not a proof):

34. log(2x) = ?

35. log(2x) = ?

36. log(2x) = log(2) + log(x)

40. Power Rule : logaMr = r logaMThink of it as repeated uses of r times

49. NEVER DO THIS • log ( x + y) = log(x) + log(y) (ERROR) • WHY is that wrong? • Log laws tell use that log(x) + log(y) = log ( xy) Not log(x + y)

50. Consider 5 = 5 You know that the and the are equal