1. Exponential & Logarithmic Functions
REVIEW
11.1 – 11.4
Exponential & Logarithmic Functions

2. Properties of ExponentsOR

3. Evaluate:
1.) 2.) 3.) 4.) 5.) 6.) 7.) 8.) 9.) 10.) 11.) 12.)

4. Express using rational exponents:
1.) 2.) 3.) 4.) 5.) 6.) 7.) 8.) 9.) 10.) 11.) 12.)

5. Express using radicals:
1.) 2.) 3.) 4.) 5.) 6.) 7.) 8.) 9.) 10.) 11.) 12.)

6. Simplify:
1.) 2.) 3.) 4.) 5.) 6.) 7.) 8.) 9.) 10.) 11.) 12.)

7. General Form of Exponential Function y = b x where b > 1
Domain: All reals
Range:
y > 0
x-intercept: None
y-intercept: (0, 1)

8. General Form of Exponential Function y = b (x + h) + k where b > 1
h moves graph left or right (opposite way)
k move graph up or down (expected way)
So y=3(x+2) + 3 moves the graph 2 units to the left and 3 units up
(0, 1) to (– 2, 4)

9. Graph: X Y -2 -1 1 2 3 9 3 1
1/3
0.111
0.037
Decreasing for all of x

10. Graph: X Y -2 -1 1 2 3
0.125
0.25
0.5 1 2 4
Increasing for all of x

11. Graph: X Y -2 -1 1 2 3
-0.5 -1 -2 -4 -8
-16
Decreasing for all of x

12. Graph: X Y -2 -1 1 2 3 8 4 2 1
0.5
0.25
Decreasing for all of x

13. Graph: X Y -2 -1 1 2 3
0.25
0.5 1 2 4 8

14. Graph: X Y -2 -1 1 2 3 4 2 1
0.5
0.25
0.125

15. Converting between Exponents & Logarithms
WRITE EACH EQUATION INTO A LOG FUNCTION:
WRITE EACH EQUATION INTO A EXPONENTIAL
FUNCTION:

16. Evaluate: 3 -3 6
Evaluate: -4 1
256

17. Properties of Logarithms (Shortcuts)
logb1 = 0 (because b0 = 1)
logbb = 1 (because b1 = b)
logb(br) = r (because br = br)
blog b M = M (because logbM = logbM)

18. Properties of Logarithms
logb(MN)= logbM + logbN
Ex: log4(15)= log45 + log43
logb(M/N)= logbM – logbN
Ex: log3(50/2)= log350 – log32
logbMr = r logbM
Ex: log7 (103) = 3 log7 10
If logbM = logbN Then M = N
log11 (1/8) = log11 8-1

19. TRY THESE TO SOLVE THESE:
First I would change to exponential form!!

20. 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.

21. Can anyone give us an explanation ?
Our final concern then is to determine why logarithms like the one below are undefined.
Can anyone give us an explanation ?
One easy explanation is to simply rewrite this logarithm in exponential form.
We’ll then see why a negative value is not permitted.
What power of 2 would gives us -8 ?
Hence expressions of this type are undefined.

22. SOLVE EACH LOGARITHM EQUATION:
1.) )
3.) )

23. Same Base Solve: 4x-2 = 64x 4x-2 = (43)x 4x-2 = 43x x–2 = 3x -2 = 2x
64 = 43
If bM = bN, then M = N
If the bases are already =, just solve the exponents

24. You Do
Solve 27x+3 = 9x-1

25. Review – Change Logs to Exponents
32 = x, x = 9
log3x = 2
logx16 = 2
log 1000 = x
x2 = 16, x = 4
10x = 1000, x = 3

26. Example 7xlog25 = 3xlog25 + ½ log225 log257x = log253x + log225 ½
7x = 3x + 1
4x = 1

27. Example 1 – Solving Simple Equations
a. 2x = 32 2x = 25 x = 5
b. ln x – ln 3 = 0 ln x = ln 3 x = 3
c = 9 3– x = 32 x = –2
d. ex = 7 ln ex = ln 7 x = ln 7
e. ln x = –3 eln x = e– 3 x = e–3
f. log x = –1 10log x = 10–1 x = 10–1 =
g. log3 x = 4 3log3 x = 34 x = 81

28. Example 6 – Solving Logarithmic Equations
a. ln x = 2
eln x = e2
x = e2
b. log3(5x – 1) = log3(x + 7)
5x – 1 = x + 7
4x = 8
x = 2

29. Example 6 – Solving Logarithmic Equations
cont’d
c. log6(3x + 14) – log6 5 = log6 2x
3x + 14 = 10x
–7x = –14
x = 2

30. Given the original principal, the annual interest rate, and the amount of time for each investment, and the type of compounded interest, find the amount at the end of the investment.
1.) P = $1,250; r = 8.5%; t = 3 years; quarterly
2.) P = $2,575; r = 6.25%; t = 5 years, 3 months; continuously