Section 4.4 Logarithmic Functions

Definition:

2) A logarithmis

merely a name for a certain exponent!

Important Result…1) The log function

and the exponential functions are inverses of each other!

xbyx yb log

1. Special Logarithms - base 10 and e

ya axxy log

2. Changing from log to exponential form

3log 1) 2 x

2log 2) 3 x

)log(2 )3 x

xln2 4) y

a axxy log

x416 5)

x5125 6)

3. Changing from exponential to log form

x10001 7)

xe51 8)

ya axxy log

1log 1) 2

16log 3) 2/1

4. Evaluating Logarithms

81

1log 2) 9

10log 4)

4e ln 5)y

a axxy log

ya axxy log

5. Special Properties

Inverse Functionsxbxf )(

Switch and solve:1) Replace f(x) with y:2) Interchange x and y:3) Solve for y:4) Replace y with

xxf blog)(1

6. Log is the Inverse of xbxf )(

)(1 xf

Inverse Property of Logarithms

xb

xbx

xb

b

log

log

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

xxf 2)(

Sketch the inverse of

reflecting graph of

over

)(log)( 21 xxf

Domain:Range:Key Points:Asymptotes:

7.Graphing the inverse of xxf 2)( xxf 2)(

xy 2

xy

Graph directlyxxf 5log)( a) Key Points

b) Domain

c) Range

d) Asymptotes

xxf 2/1log)(

a) Key Points

b) Domain

c) Range

d) Asymptotes

is the inverse of

8 Two Methods for Graphing Log Functions

Method 1: Directly

Method 2: Use Inverse

1)2(log)( 2 xxf

a) Key Points of parent

b) Transformations

c) Domain

d)Range

e) Asymptotes

Method 1: Determine the graph of a log function, using transformations of the parent function.

1)2(log)( 2 xxf

1) Find 2) Graph3) Reflect over y = x to graph

)(1 xf

)(1 xf

)(xf

Method 2: Determine the graph of a log function by graphing its inverse function and reflect over y=x

9. Domain of a logarithmic function

Determine the domain for these functions.

1ln)(x

xxf

12log)( 4 xxf

10. Change-of-Base Formula

log lnlog

log lnaM M

Ma a

Example.

Find an approximation for )5(log2

11. Solving: Log = Constant

Logarithmic Equations

Solve for variable inside the log expression.

15)82(log 4 x

Use the definition: xC blog xbC

12. Solving: Exponential = Constant

Solve for variable in the exponent

Use the definition: Cx blogCb x

723 1 x

11. Solving: Log = Log

If then M = N NM aa loglog

)12(loglog 1) 33 xx

Solve for variable inside log on each side.

When solving log functions, we must

check that a solution lies in the

domain!

Summary: Inverse Properties of Logarithmic and Exponential Functions

The Logarithmic and Exponential Functions are inverses of each other.

Example of the relationship: Let xxfxg x2log)( and 2)(

Inverse Property of xxfbxg bx log)( and )(

xbxgfbxfg xb

xb log))(( andx ))(( log

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

xxf 3)(

Exponential functions and log functions are inverse functions

of each other.

xxf 31 log)(

Domain:Range:Key Points:Asymptotes:

Graphing Logarithmic Functions