2x = 4 2y = 8 2z = 6X = 2 y = 3 Z = ?

Mathematicians use a LOGARITHM to find z and we will study logarithmic functions this unit

A logarithm is the inverse of an ______________ function Exponential

Inverses

These 2 graphs are reflections over the line _________

X Y

-3

-2

-1

0

1

2

3

X Y

-3

-2

-1

0

1

2

3

y = x

1/8

1/4

1/2

1

2

4

8

1/8

1/4

1/2

1

2

4

8

Note exp fcn has H.A. and log fcn has

V.A.

If x = b y

then ________Look at the log function graph:

x > 0

log b x = y

always greater than 0x is ____________________

domain is _________Value of asymptote

If x = b y

then ________log b x = y

Convert the Exponential Equations to Logarithms

1. 2. 3. 32 = 94216 210100

log 2 16 = 4 log 10 100 = 2 log 3 9 = 2

Note that we are changing form ….

not solving

If x = b y

then ________ log b x = y

Convert the Exponential Equations to Logarithms

4. 5. 6. 1 = 50

16

14 2

1.010 1

log 4 = -2 log 10 0.1 = -1 log 5 1 = 016

1

If x = b y

then ________log b x = y

Write the Logarithmic Equations in Exponential Form

7. 8. 9.

6482 823

log 8 64 = 2 log 2 8 = 3 log 100 = 2

100102 When no base is written ….it is a common log with base 10

If x = b y

then ________y = log b x

Evaluate each Logarithm

1. 2. =x 3. log1000 = x

273 x6

16 x

log3 27 = x log6

100010 x6

1

333 x

3x166 x

1x

31010 x

3x

Now we are solving for

x

If x = b y

then ________y = log b x

Evaluate each Logarithm

4. 5. =x 6. log816 =x

279 x8

1

2

1

x

log 9 27 = x log½

168 x8

1

32 33 x

32 x

2

3x

31 22 x

31 x

43 22 x

43 x

3x 3

4x

Special Logarithm Values

logb1=_____ logbb=_____ logbbx=_____

b x = 1

0

b x = b

1

b x = b x

x

Why are these good rules to know: (not on your notes)

3)7(log7 xyFind the y-intercept of

37log7 y31y

Substitute 0 for x(0,4)

For example:

log x = _____________ (The log key on the calc. is the common log)

10

log10x

Use the change of

base Formula:

log b x =

b

x

b

x

10

10

log

log

log

log

Example:

log 2 7 =

807.22log

7log

62 z

z6log2 z2log

6log3log585.2 notz

xy blog

Vertical Shift:

1a 10 a

0a

khxay b )(log

Parent Function:

Horizontal Shift:

Stretch/Compress:

Reflection in x-axis:

The k

khxay b )(log The h

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.

xy 2

y = log 2 x

x = log 2 ySwitch variables to find inverse equations

Convert from log to exp. form

4)1(log2 xy

Look above at the parent function of y = log2x

Left 1Horiz shift ________

Vert Shift = _______

V Asymptote: ______Domain: __________

Up 4

x = -1

ParentFunctionY=log2x

X Y1/4 -21/2 -1

1 0

2 1

4 2

8 3

4

1

X-intercept:______

0 = log 2 (x+1) +4

– 4 = log 2 (x+1)

2 -4 = x+1

x 116

1

16

15x

)0,16

15(

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=16B. log216=4C. log416=2D. log164=2

A. logbc=aB. logcb=aC. logab=cD. logac=b

A. bc=aB. ac=bC. ab=cD. ba=c

A. 39=2B. 23=9C. 32=9D. 92=3

A. 3B. 4C. 16D. 256

A. -4B. -3C. 3D. 4

2

1

A. -4B. -27C. 27D. 243

3

1

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

xxf 3log)(

A. X 1

B. X 1

C. X -1

D. X -1

A. (7,0)

B. (8,0)

C. (9,0)

D. (10,0)

A. X=1/3

B. X=27

C. X=-2

D. X=-27

9

13 x

A. X=-5

B. X=-3

C. X=3

D. X=7

16

12 1 x

A. X=1.5

B. X=5

C. X=6

D. X=9

2273 x

A. X=10/3

B. X=4

C. X=16

D. X=64

28 322 xx

A. X=-81

B. X=9

C. X=2/3

D. X=3/2

27

83 x

8

27

1

3

x

33

2

3

x

2

3x

A. X=-27

B. X=-9

C. X=-4

D. X=27

813

1

x

4x

41 33 x

41 x

Like HW 8.2