TABLE OF CONTENTS

3.3         The Natural Logarithmic Function

 

Goals:

·         Apply knowledge of logarithms to base e

 

Prerequisite skills and knowledge:

 

·         a working knowledge of logarithms to the base 10

·         a working knowledge of basic properties of the natural exponential function

 

 

Terms to know:

 

·         continuous growth or decay

·         e

·         exponential function
  •   natural exponential
  •   logarithm

 

 

Concept Prep:  Review of natural exponential

 

Recall the number e from our work with exponential functions.   The summary box is reproduced below for your convenience.

 

e   2.7

e is an irrational number

When used as a base for an exponential function, the function is called the natural exponential  function.

 

This natural exponential function is used when modeling continuous growth or decay.

 

 

 

            Recall from section 3.1 that in the equation , , A gives us the amount of money we would have if we invest $1000 in an account that compounds interest continuously at a nominal rate of 6%.  Suppose now that we need $3000 for a down payment on a car.  To simplify our work even more, let’s suppose also that some generous soul will give us 100% interest rather then 6%.   Our equation becomes

 

 

How could we find the value of t?

 

If we divide both sides by 1000, we now have

 

  (1)

 

We want the exponent to which we would take e to get 3.  

 

You’ll spend time in your Algebra for Calculus solving these types of equations.  For now, we want you to understand that the number e can be used as a base for a logarithm.  You would expect that such a logarithm would be written , but we usually write it  refer to it as the “natural logarithm.”

 

 

 

The notation , which means  means the power you raise e to in order to get x.  Logarithms to the base e are called natural logarithms.

 

 

 

 

Example 1.   Express each equation in exponential form.

 

                                   

 

SOLUTION

 

a)  Since the base is e and the logarithm is an exponent, we write:      

 

b)  Since the base is e and the logarithm is an exponent, we write:      

 

c)  Since the base is e and the logarithm is an exponent, we write:     

 

d)  Since the base is e and the logarithm is an exponent, we write:     

 

 

Example 2. Express each equation in logarithmic form:

 

 

 

 

SOLUTION

 

a)   Since the base is e we use ln notation.  The logarithm is

      the exponent.  We mean , but we write:                                 

 

b)   Since the base is e we use ln notation.  The logarithm is

      the exponent.  We mean , but we write:                                   

               

c)   Since the base is e we use ln notation.  The logarithm is

      the exponent.  We mean , but we write:                           

 

d)  Since the base is e we use ln notation.  The logarithm is

      the exponent.  We mean , but we write:                                 

 

 

Example 3.  Evaluate each of the following without the use of your calculator.

 

 

 

SOLUTION

 

a)       

            Remember that we can think of  as meaning .

            Since a logarithm is an exponent and the base is e,

           we want the power to which we take e that will give us 1:                               

           

            Since any number to the 0 power is 1, the logarithm is 0:         

 

b)   

            Remember that we can think of  as meaning  

            Since a logarithm is an exponent and the base is e,

           we want the power to which we take e that will give us e:                              

           

            Since any number to the 1st power is itself the logarithm

           is 1:                                                                                                    

 

c)    

            Remember that we can think of  as meaning  

            Since a logarithm is an exponent and the base is e,

           we want the power to which we take e that will give us e3:                                        

            Silly question.  The answer is given in the problem!                     

                                   

d)   

            Remember that we can think of   as meaning  

           Since a logarithm is an exponent and the base is e,

           we want the power to which we take e that will give us .         

           Recall that , so we are really looking for

            the power to which we take e to get .                          

 

 

 

      Checkpoint: The Natural Log Function

 

More worked examples

 

Homework problems

 

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