Logarithmic Logarithmic FunctionsFunctions

ObjectiveObjective

To graph logarithmic functionsTo graph logarithmic functions To evaluate logatrithmsTo evaluate logatrithms

The inverse function of an exponential The inverse function of an exponential function is the logarithmic function.function is the logarithmic function.

For all For all positivepositive real numbers x and b, real numbers x and b, b>0 and bb>0 and b1, y=log1, y=logbbx if and only if x=bx if and only if x=byy..

The domain of the logarithmic function is The domain of the logarithmic function is

( ) logbf x x

• 0, .

The range of the function is The range of the function is Since the log function is the inverse Since the log function is the inverse

of the exponential function, their of the exponential function, their graphs are symmetric with respect to graphs are symmetric with respect to the line y=x.the line y=x.

, .

2xy

2logy x

y x

**Remember that since the log function is the inverse of theexponential function, we can simply swap the x and y values of our important points!

1( 1, )

(0,1)

(1, )

b

b

1 , 1

1,0

,1

b

b

12 , 1

1,0

2,1

2logy x

xy b logby x

•Logarithmic Functions where b>1 are increasing, one-to-one functions.

•Logarithmic Functions where 0<b<1 are decreasing, one-to-one functions.

•The parent form of the graph has an x-intercept at (1,0) and passes through (b,1) and

logbf x x

1( , 1)b

There is a vertical asymptote There is a vertical asymptote at x=0.at x=0.

The value of b determines the The value of b determines the flatness of the curve.flatness of the curve.

The function is neither even The function is neither even nor odd. There is no nor odd. There is no symmetry.symmetry.

There is no local extrema. There is no local extrema.

More Characteristics of logbf x x

• The domain is The domain is • The range isThe range is• End Behavior: End Behavior: • AsAs• As As • The x-intercept is The x-intercept is • The vertical asymptote The vertical asymptote

isis

•

0 , ( ) .x f x , ( ) .x f x

, . 0, .

1,0 .0.x

• There is no y-intercept.There is no y-intercept.• There are no There are no

horizontal asymptotes.horizontal asymptotes.• This is a continuous, This is a continuous,

increasing function.increasing function.• It is concave down.It is concave down.

Graph: 3( ) logf x x

3logy x3y x

Important Points:

13 , 1 1,0

3,1

Domain: Range: x-intercept:

VerticalAsymptote:

Inc/dec?

Concavity?

, 0,

1,0

0x

increasing

down

0 , ( ) .x f x

, ( ) .x f x

Graph: 12

logy x

Important Points: 2, 1 1,0 12 ,1

Domain: Range: x-intercept:

VerticalAsymptote: Inc/dec?Concavity?

0, ,

1,0

0x decreasing

up

*Reflects @ x-axis.

Transformations2log 1y x 1

22log 3y x 3log ( 1) 2y x

Vertical stretch of 2.Vertical shift up 1.Reflect @ x-axis.

Vertical shift down 3.

Horizontal shift right 1.Vertical shift up 2.

Domain: Range: x-intercept:

VerticalAsymptote:

Inc/dec?Concavity?

0, ,

2, 0

0x decreasing

up

* Reflect @ x-axis.

VerticalAsymptote: Inc/dec?Concavity?

VerticalAsymptote:

Inc/dec?Concavity?

0, ,

1, ,

1.111,0(.354,0)

1x 0x decreasing increasing

downup

Domain: Range: x-intercept:

Domain: Range: x-intercept:

More TransformationsMore Transformations3log 2y x

Horizontal shrink ½ .

3log (2 1)y x Horizontal shrink ½ .Horizontal shift right ½ ..

Domain: Range: x-intercept: Vertical

Asymptote:

Inc/dec?

Concavity?

0, ,

12 , 0

0x increasing

down

Domain: Range: x-intercept: Vertical

Asymptote:

Inc/dec?

Concavity? down

increasing

12x

1,0

, 1

2 ,

logbh x ax c The asymptote of a logarithmic function of thisform is the line To find an x-intercept in this form, let y=o inthe equation To find a y-intercept in this form, let x=o inthe equation

0.ax c

log( ).y ax c

log( ).y ax c

3log (2 1)y x 2 1 0x

12x

12x is thevertical asymptote.

3

0

0 log (2 1)

3 2 1

x

x

1 2 1x 2 2x

1x 1,0 is the x-intercept

3log (2(0) 1)y

3log ( 1)y Since this is notpossible, there isNo y-intercept.

Check it out!

3log (2 1)y x Horizontal shrink ½ .Horizontal shift right 1.

Domain: Range: x-intercept: Vertical

Asymptote:

Inc/dec?Concavity?

12 ,

, 1,0

12x

increasingdown

12x is the vertical asymptote.

1,0 is the x-intercept.

Since this is notpossible, there isNo y-intercept.

3log ( 1)y

Common Log & Natural Common Log & Natural LogLog

• A logarithmic function with base 10 is called a Common Log.

Denoted: • A logarithmic function with base e is called a Natural Log.

Denoted:

logf x x

lnf x x

*Note there is no base written.

TransformationsCommon Log

log 3 10h x x

3 10 0x to find the V.A.103x is the V.A.

0 log 3 10x

Let

Let 0y to find the x-intercept.

010 3 10x 1 3 10x 3 x 3,0 is the x-intercept.

Let 0x to find the y-intercept.

log(3(0) 10)y

10 10y 1y

0,1 is the y-intercept.

Horizontal shrink 1/3.Horizontal shift left 10/3.

103log 3( )h x x

Domain:Range:Inc/Dec:Concavity:

103 ,

, increasing

down

TransformationsNatural Log

2ln 4 8 2h x x Let 4 8 0x to find the V.A.

2x is the V.A.0y to find the x-intercept. 0 2ln 4 8 2x

2 2ln(4 8)x 1 ln(4 8)x 1 4 8e x

Let

2.718 4 8x 1.3205x

1.3205,0 is the x-intercept.

Reflect @ x-axis.Vertical stretch of 2.Horizontal shrink of ¼ .Horizontal shift left 2.Vertical shift up 2.

Let 0x to find the y-intercept.

2ln 4(0) 8 2y 2ln8 2y

22 8y

e

2.159y

0, 2.159 is the y-intercept.

2ln 4( 2) 2h x x

Inc/dec?Concavity?

decreasingup

Domain: Range: x-intercept:

2,

, 1.321,0

Change of Base FormulaChange of Base Formula

loglog

logb

xx

b

Use this formula for entering logs with bases other than 10 or ein your graphing calculator.So, if you wanted to graph , you would enter in your calculator.

3log 1y x log( 1)

log3

xy

Either the natural or common log may be used in the changeof base formula. So, you could also enter inYour calculator.

ln( 1)

ln3

xy