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Exponents

Exponential Equations I

Exponential Functions

The Exponential Function Base e

The Logarithm Function

Logarithmic & Exponential Equations II

Applications:Growth & Decay

Review&Test

 

 UNIT 4  : EXPONENTIAL & LOGARITHMIC FUNCTIONS

 LESSON 5: THE LOGARITHMIC FUNCTION

 

Logarithmic Notation:

The exponential statement  23 = 8  may be rewritten in what is called logarithmic form

as follows:   log28 = 3  which reads:  the logarithm of 8, base 2, equals 3. 

Any exponential statement may be rewritten in this form.

 

Example 1:

Complete the following chart  by converting between exponential and logarithmic form.

                       

           

Exponential Form

Logarithmic Form

34 = 81

52 = 25

log232 = 5

log101000 = 3

 

Solution:

 

                       

           

Exponential Form

Logarithmic Form

34 = 81

log381 = 4

52 = 25

log525 = 2

25 = 32

log232 = 5

103 = 1000

log101000 = 3

 

 

 

Note:  When we read  log28, we ask the question  “To what exponent must base 2 be raised to give 8?”  The answer

is of course 3 and this idea gives rise to the following definition.

 

Definition:  the expression  logax  is defined to mean  “ the exponent to which base a must be raised

to give x.  The expression reads:  “ the logarithm of x, base a.”

 

Hence  log10100 means the exponent to which base 10 must be raised to give 100.  The answer is 2, giving the

statement  log10100 = 2.

 

Hence  log381 means the exponent to which base 3 must be raised to give 81.  The answer is 4, giving the

statement  log381 = 4.

 

 

The Logarithmic Function:

Logarithmic functions are the inverses of exponential functions.  How does this come about??  See example below.

 

 

Table of Values & Graphs:          

 

 

x

y

 

-3

 

-2

 

-1

0

1

1

2

3

8

4

16

 

 

 

Now form the inverse of this function by interchanging x and y in the ordered pairs and the equation.

 

 

 

 

 

x

y

-3

-2

-1

1

0

1

2

8

3

16

4

             

 

 

 

 

The equation of the inverse is found by interchanging x and y in the equation  y = 2x.  This yields  x = 2y.  If we put this

equation in logarithmic notation,   we obtain the logarithmic function   y = log2x. 

The graph is a reflection in the line y = x of the exponential function y = 2x and is shown in red above.

 

 

 

 

 

 

 

Table of Values & graph:

 

x

y

 

-3

 

-2

-1

3

0

1

 

1

 

2

 

3

 

 

 

 

 

 

 

Now form the inverse of this function by interchanging x and y in the ordered pairs and the equation.

 

 

 

x

y

-3

-2

3

-1

1

0

1

2

3

 

 

 

 

The equation of the inverse is found by interchanging x and y in the equation  y = (1/3)x.  This yields  x = (1/3)y.  If we put this

equation in logarithmic notation,  we obtain the logarithmic function   y = log(1/3)x. 

The graph is a reflection in the line y = x of the exponential function y = (1/3)x and is shown in red above.

 

 

The Natural Logarithmic Function  y = logex  or  y = ln x:

 

 

x

 

-3

 

-2

 

-1

  0

1

1

2

3

4

 

Now form the inverse of this function by interchanging x and y in the ordered pairs above.

 

 

 

 

x

y

-3

-2

 

-1

1

0

1

2

5

4

 

 

 

 

 

 

 

 

The equation of the inverse is found by interchanging x and y in the equation  y = ex.  This yields  x = ey.  If we put this

equation in logarithmic notation,  we obtain the logarithmic function   y = logex or y = lnx. 

The graph is a reflection in the line y = x of the exponential function y = ex and is shown in red above.

 

 

 

 

 

 

    

                                                                                                                                                                                

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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