1. Notes Day PAP Algebra 2 Objective: TLW… develop the definition of logarithms by exploring and describing the relationship between exponential functions and their inverses (2A.11.A) Use parent functions to describe effects on graphs of exponential and logarithmic functions (2A.11.B) Examine asymptotic behavior (2A.11.B) Solve exponential/logarithmic equations using algebraic methods (2A.11.D)

2. 2x = 4 2y = 8 2z = 6 Find x, y, and z Z = ? X = 2 y = 3 Exponential
Mathematicians use a LOGARITHM to find z and we will study logarithmic functions this unit
Exponential
A logarithm is the inverse of an ______________ function

3. Note exp fcn has H.A. and log fcn has V.A.
f(x) = 2x
f-1x = log2x
Inverses X Y -3 -2 -1 1 2 3 X Y -3 -2 -1 1 2 3
1/8
1/8
Note exp fcn has H.A. and log fcn has V.A.
1/4
1/4
1/2
1/2 1 1 2 2 4 4 8 8
These 2 graphs are reflections over the line _________
y = x

4. Convert Exponentials and Logarithms
If x = b y
then ________
log b x = y
Look at the log function graph:
x is ____________________
domain is _________
always greater than 0
Value of asymptote
x > 0

5. Note that we are changing form ….
Convert the Exponential Equations to Logarithms
If x = b y
then ________
log b x = y
= 9
log 3 9 = 2
log 2 16 = 4
log = 2
Note that we are changing form ….
not solving

6. When no base is written ….it is a common log with base 10
Write the Logarithmic Equations in Exponential Form
If x = b y
then ________
log b x = y
log 8 64 = 2
log 100 = 2
log 2 8 = 3
When no base is written ….it is a common log with base 10

7. If x = b y then ________ y = log b x 7. 8. =x 9. log1000 = x
Evaluate each Logarithm
Now we are solving for x
If x = b y
then ________
y = log b x
=x log1000 = x
log = x
log6

8. Special Logarithm Values1 x
logb1=_____
logbb=_____
logbbx=_____
b x = 1
b x = b
b x = b x
Why are these good rules to know: (not on your notes)
Find the y-intercept of
Substitute 0 for x
(0,4)

9. A common logarithm is a logarithm that uses base ______10
For example:
log x = _____________ (The log key on the
calc. is the common log)
log10x

10. Use the change of base Formula: log b x =
change of base formula Remember on the first slide when we wanted to solve
Use the change of
base Formula:
log b x =
Example:
log 2 7 =
So now solve

11. Translating Logarithmic Functions
Parent Function:
The k
Vertical Shift:
The h
Horizontal Shift:
Stretch/Compress:
Reflection in x-axis:

12. This current slide is not in your notes – but lets prove why
On an earlier slide we graphed an exponential function and its inverse.
This current slide is not in your notes – but lets prove why
y=2x and y = log 2x are inverses.
y = log 2 x
Switch variables to find inverse equations
x = log 2 y
Convert from log to exp. form

13. Graph Look above at the parent function of y = log2x X Y -2 -1 1 2 4 8
1/4 -2
1/2 -1 1 2 4 8 3
Horiz shift ________
Vert Shift = _______
V Asymptote: ______
Domain: __________
Left 1 4
Up 4
x = -1
x > -1 1
X-intercept:______
0 = log 2 (x+1) +4
– 4 = log 2 (x+1)
2 -4 = x+1

14. Activity:
Now lets see what you know. I will show you some problems. When I ask for the answer, please show the color of the matching correct answer.
HW : WS 8.2 – which is is due next class. We will also be taking a quiz next class on these concepts.

15. Express 24=16 in Logarithm Form.
A. log24=16
B. log216=4
C. log416=2
D. log164=2

16. Express ab=c in Logarithm Form.
A. logbc=a
B. logcb=a
C. logab=c
D. logac=b

17. Express logab=c in Exponential Form.
A. bc=a
B. ac=b
C. ab=c
D. ba=c

18. Express log39=2 in Exponential Form.
B. 23=9
C. 32=9
D. 92=3

19. Evaluate log28
A. 3
B. 4
C. 16
D. 256

20. Evaluate log (8)
A. -4
B. -3
C. 3
D. 4

21. Evaluate log (81)
A. -4
B. -27
C. 27
D. 243

22. How is the graph of f(x) = log3(x-1)-5 transformed from its parent function
A. Translated down 1 and left 5 B. Translated up 1 and left 5 C. Translated left 1 and down 5 D. Translated right 1 and down 5

23. What is the domain of the graph f(x) = log3(x-1)-5
A. X 1 B. X 1 C. X -1 D. X -1

24. What is the x-intercept of the graph f(x) = log3(x-1)-2
A. (7,0) B. (8,0) C. (9,0) D. (10,0)

25. Solve for x:
A. X=1/3 B. X=27 C. X=-2 D. X=-27

26. Solve for x:
A. X=-5 B. X=-3 C. X=3 D. X=7

27. Solve for x:
A. X=1.5 B. X=5 C. X=6 D. X=9

28. Solve for x:
A. X=10/3 B. X=4 C. X=16 D. X=64

29. Solve for x:
A. X=-81 B. X=9 C. X=2/3 D. X=3/2

30. Solve for x:
A. X=-27 B. X=-9 C. X=-4 D. X=27

31. Quiz Next Class:
Like HW 8.2