1. 4.3 Laws of Exponents
Romans 8:24-25 For in this hope we were saved.
Now hope that is seen is not hope. For who hopes for
what he sees? But if we hope for what we do not
see, we wait for it with patience.

2. Remember that logs are exponents.
Therefore, these laws should make sense ...

3. Remember that logs are exponents.
Therefore, these laws should make sense ...
1) log a (AB) = log a A + log a B

4. Remember that logs are exponents.
Therefore, these laws should make sense ...
1) log a (AB) = log a A + log a B
⎛ A ⎞
2) log a ⎜ ⎟ = log a A − log a B
⎝ B ⎠

5. Remember that logs are exponents.
Therefore, these laws should make sense ...
1) log a (AB) = log a A + log a B
⎛ A ⎞
2) log a ⎜ ⎟ = log a A − log a B
⎝ B ⎠
3)
c
log a (A) = c log a A

6. Logs can be manipulated using these laws
to either expanded form or combined form.

7. Expand
1) log 3 (9x)

8. Expand
1) log 3 (9x)
log 3 9 + log 3 x

9. Expand
1) log 3 (9x)
log 3 9 + log 3 x
2 + log 3 x

10. Expand
2) log 7 (x y )
3 4

11. Expand
2) log 7 (x y )
3 4
3 4
log 7 x + log 7 y

12. Expand
2) log 7 (x y )
3 4
3 4
log 7 x + log 7 y
3log 7 x + 4 log 7 y

13. Expand
⎛ a b ⎞
3) ln ⎜ 8 ⎟
⎝ c ⎠

14. Expand
⎛ a b ⎞
3) ln ⎜ 8 ⎟
⎝ c ⎠
1
⎛ ⎞
ln ⎜ ab ⎟ − ln ( c )
2 8
⎝ ⎠

15. Expand
⎛ a b ⎞
3) ln ⎜ 8 ⎟
⎝ c ⎠
1
⎛ ⎞
ln ⎜ ab ⎟ − ln ( c )
2 8
⎝ ⎠
1
8
ln a + lnb − ln c
2

16. Expand
⎛ a b ⎞
3) ln ⎜ 8 ⎟
⎝ c ⎠
1
⎛ ⎞
ln ⎜ ab ⎟ − ln ( c )
2 8
⎝ ⎠
1
8
ln a + lnb − ln c
2
1
ln a + lnb − 8 ln c
2

17. Combine
3
4) 3ln x + ln ( x + 3)
2

18. Combine
3
4) 3ln x + ln ( x + 3)
2
3
3
ln x + ln ( x + 3) 2

19. Combine
3
4) 3ln x + ln ( x + 3)
2
3
3
ln x + ln ( x + 3) 2
3
⎛ 3 ⎞
ln ⎜ x ( x + 3) ⎟
2
⎝ ⎠

20. Combine
3
4) 3ln x + ln ( x + 3)
2
3
3
ln x + ln ( x + 3) 2
3
⎛ 3 ⎞
ln ⎜ x ( x + 3) ⎟
2
⎝ ⎠
Book answer could be: (
ln x 3
( x + 3) 3
)

21. Combine
5) log x − 3log y + log ( x + y )

22. Combine
5) log x − 3log y + log ( x + y )
order of operations ... from left to right

23. Combine
5) log x − 3log y + log ( x + y )
order of operations ... from left to right
⎛ x ⎞
log ⎜ 3 ( x + y )⎟
⎝ y ⎠

24. Combine
3 1
6) ln (16y ) − 2 ln x + ln y
4 4

25. Combine
3 1
6) ln (16y ) − 2 ln x + ln y
4 4
3 1
2
ln (16y ) − ln x + ln y
4 4

26. Combine
3 1
6) ln (16y ) − 2 ln x + ln y
4 4
3 1
2
ln (16y ) − ln x + ln y
4 4
3 1
⎛ ⎞ 2
ln ⎜ 8y ⎟ + ln y − ln x
4 4
⎝ ⎠

27. Combine
3 1
6) ln (16y ) − 2 ln x + ln y
4 4
3 1
2
ln (16y ) − ln x + ln y
4 4
3 1
⎛ ⎞ 2
ln ⎜ 8y ⎟ + ln y − ln x
4 4
⎝ ⎠
3 1
⎛ ⎞ 2
ln ⎜ 8y y ⎟ − ln x
4 4
⎝ ⎠

28. Combine
3 1
6) ln (16y ) − 2 ln x + ln y
4 4
3 1
2
ln (16y ) − ln x + ln y
4 4
3 1
⎛ ⎞ 2
ln ⎜ 8y ⎟ + ln y − ln x
4 4
⎝ ⎠
3 1
⎛ ⎞ 2
ln ⎜ 8y y ⎟ − ln x
4 4
⎝ ⎠
8y
ln 2
x

29. Remember: Change of Base Formula

30. Remember: Change of Base Formula
log x ln x
log b x = =
logb lnb

31. True or False? (Group Consensus)
calculator ok to use ...

32. True or False? (Group Consensus)
calculator ok to use ...
1) log10 ( 3 + 5 ) = log10 3 + log10 5
7 log 7
2) log =
2 log 2
3) 4 ln 6 = ln 6 4
4
4) 4 ln 6 = ( ln 6 )

33. True or False? (Group Consensus)
calculator ok to use ...
1) log10 ( 3 + 5 ) = log10 3 + log10 5 False
7 log 7
2) log =
2 log 2
3) 4 ln 6 = ln 6 4
4
4) 4 ln 6 = ( ln 6 )

34. True or False? (Group Consensus)
calculator ok to use ...
1) log10 ( 3 + 5 ) = log10 3 + log10 5 False
7 log 7
2) log = False
2 log 2
3) 4 ln 6 = ln 6 4
4
4) 4 ln 6 = ( ln 6 )

35. True or False? (Group Consensus)
calculator ok to use ...
1) log10 ( 3 + 5 ) = log10 3 + log10 5 False
7 log 7
2) log = False
2 log 2
3) 4 ln 6 = ln 6 4
True
4
4) 4 ln 6 = ( ln 6 )

36. True or False? (Group Consensus)
calculator ok to use ...
1) log10 ( 3 + 5 ) = log10 3 + log10 5 False
7 log 7
2) log = False
2 log 2
3) 4 ln 6 = ln 6 4
True
4
4) 4 ln 6 = ( ln 6 ) False

37. HW #6
I’ve got a theory that if you give 100% all of the time,
somehow things will work out in the end.
Larry Bird