x
2
=4
X=2
y
2
=8
y=3
z
2
=6
Z=?
Mathematicians use a LOGARITHM to find z
and we will study logarithmic functions this unit
A logarithm is the inverse of an ______________
Exponential function
X
Inverses
Y
X
Y
-3 1/8
1/8 -3
-2 1/4
1/4 -2
-1 1/2
0
1
1
2
2
4
3
8
Note
exp fcn
has H.A.
and log
fcn has
V.A.
1/2 -1
1
0
2
1
4
2
8
3
These 2 graphs are reflections over the line _________
y=x
If x = b y
log b x = y
then ________
Look at the log function graph:
x is ____________________
always greater than 0
domain is _________
x>0
Value of
asymptote
Convert the Exponential Equations to Logarithms
If x =
y
b
log b x = y
then ________
1. 16  2
4
log 2 16 = 4
2.
100  10
2
log 10 100 = 2
3. 32 = 9
log 3 9 = 2
Note that we are
changing form ….
not solving
Convert the Exponential Equations to Logarithms
If x =
y
b
log b x = y
then ________
4. 4
log
2
1
4 16
1

16
= -2
5. 10
1
 0.1
log 10 0.1 = -1
6. 1 = 50
log 5 1 = 0
Write the Logarithmic Equations in Exponential Form
If x =
y
b
log b x = y
then ________
7. log 8 64 = 2
8  64
2
8. log 2 8 = 3
2 8
3
9. log 100 = 2
When no
base is
written
….it is a
common
log with
base 10
10  100
2
Evaluate each Logarithm
If x =
y
b
Now we are
solving for
x
y = log b x
then ________
1. log3 27 = x
3  27
x
3 3
x
x3
3
2.
1
log6 6
=x
10 x  1000
1
6 
6
x
6 6
x
3. log1000 = x
1
x  1
10  10
x
x3
3
Evaluate each Logarithm
If x =
y
b
y = log b x
then ________
4. log 9 27 = x
9  27
x
3 3
2x  3
3
x
2
2x
3
5. log½
1
8
=x 6. log816 =x
8 x  16
x
1
1

2
8
2
1x
2
3
1x  3
x3
2 2
3x
4
3x  4
4
x
3
Special Logarithm Values
logb1=_____
0
b x= 1
1
logbb=_____
b x= b
x
logbbx=_____
b x= b x
Why are these good rules to know: (not on your notes)
Find the y-intercept of
(0,4)
y  log 7 ( x  7)  3
Substitute 0 for x
y  log 7 7  3
y  1 3
10
For example:
log10x (The log key on the
log x = _____________
calc. is the common log)
2 6
z
Use the change of
base Formula:
Example:
log x log 10 x
log b x =

log b log 10 b
log 7
 2.807
log 2 7 =
log 2
log 2 6  z
log 6
z
log 2
z  2.585 not log 3
Parent Function:
y  log b x
Vertical Shift: y  a log b ( x  h)  k
The k
Horizontal Shift: y  a log b ( x  h)  k
The h
Stretch/Compress:
Reflection in x-axis:
a 1
a0
0  a 1
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
x = log 2 y
y2
x
Switch variables to
find inverse equations
Convert from log to
exp. form
y  log 2 ( x  1)  4
Look above at the parent function of y = log2x
4
1
1
1  x
16
 15
x
16
Parent
Function
Y=log2x
Horiz shift ________
Left 1
X
Y
Vert Shift = _______
Up 4
1/4
-2
1/2
-1
V Asymptote: ______
x = -1
Domain: __________
x > -1
1
0
 15
,0)
16
X-intercept:______
2
1
4
2
0 = log 2 (x+1) +4
8
3
(
– 4 = log 2 (x+1)
2 -4 = x+1
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.
A.
log24=16
B. log216=4
C. log416=2
D. log164=2
A.
logbc=a
B. logcb=a
C. logab=c
D. logac=b
A.
c
b =a
B.
c
a =b
D.
a
b =c
C.
b
a =c
A.
9
3 =2
B.
3
2 =9
D.
2
9 =3
C.
2
3 =9
A.
3
B. 4
C. 16
D. 256
1
2
A.
-4
B. -3
C. 3
D. 4
1
3
A.
-4
B. -27
C. 27
D. 243
f ( x)  log 3 x
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
A.
B.
C.
D.
X 1
X1
X -1
X  -1
A.
B.
C.
D.
(7,0)
(8,0)
(9,0)
(10,0)
1
3 
9
x
A.
B.
C.
D.
X=1/3
X=27
X=-2
X=-27
2
A.
B.
C.
D.
X=-5
X=-3
X=3
X=7
x 1
1

16
3  27
x
A.
B.
C.
D.
X=1.5
X=5
X=6
X=9
2
2
A.
B.
C.
D.
X=10/3
X=4
X=16
X=64
8x
 32
x2
x
A.
B.
C.
D.
X=-81
X=9
X=2/3
X=3/2
3
3
8

27
x
27

1
8
3
x    
2
3
3
x
2
3
A.
B.
C.
D.
1
 
3
X=-27
X=-9
X=-4
X=27
x
 81
3 
1 x
3
1x  4
x  4
4
Like
HW 8.2