An Approach to Mathematic Functions Basics

by Frank C. Fung ( 1st published in December, 2014. )

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Section IV - Defining Natural Logarithms

Summary for the Section :

• In this Section , we shall take a quick look at [ Natural Logarithms ] and its defining function ,
  • namely :
    • the [ Natural-LOG Function ] .

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The Defining Criteria for the [ Natural-LOG Function ] :

• Let us bring-in again our findings on the [ Logarithmic Class of Functions ] from the previous Section :
  • The general equation for a Logarithmic Function , denoted as the [ PHI-of-(x) ] herein , is :
    • 

      where :

      • 

    And its first-derivative is :

    • 

• For the [ Natural-LOG Function ] , we simply set the value of [ K ] equal to [ 1 ] ,
  • i.e. :
    • K = 1 .

  Thus , the resultant [ Natural-LOG Function ] , denoted as the [ LOG-of-(x) ] , is then :

  • 

  And its first-derivative is then :

  • 

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A Graphic Representation of the [ Natural-LOG Function ] :

• Let us now take a quick look at the graphic presentation of the [ Natural-LOG Function ] :
  • the diagram on-the-left below then shows the First-Derivative of the [ Natural-LOG Function ] , and

  • the diagram on-the-right below then shows the [ Natural-LOG Function ] itself .

  

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A Quick Familiarization with Natural Logarithms :

• For a slightly better understanding of the [ Natural-LOG Function ] ,
  • let us now take a quick look at the [ Natural Logarithms ] for 3 specific values of [ x ] , namely :
    • for [ x = 2 ] :
      • [ Log 2 ] = 0.69314718....

    • for [ x = 3 ] :
      • [ Log 3 ] = 1.09861228....

    • for [ x = 4 ] :
      • [ Log 4 ] = 1.38629436....

• And we recall , at this point , that :
  • 

  And that-is-to-say :

  • the [ Natural Logarithm of (x) ] is given by :
    • the area under the [ hyperbola Y = 1/X ]

      from [ 1 ] to [ x ] .

  Thus :

  • For [ Log 2 ] = 0.693147 , we then have this set of 2 diagrams below with :
    • the diagram on-the-left showing the first-derivative and

    • the diagram on-the-right showing the [ Natural-LOG Function ] itself .

  

  • For [ Log 3 ] = 1.098612 , we then have this set of 2 diagrams below with :
    • the diagram on-the-left showing the first-derivative and

    • the diagram on-the-right showing the [ Natural-LOG Function ] itself .

  

  • For [ Log 4 ] = 1.386294 , we then have this set of 2 diagrams below with :
    • the diagram on-the-left showing the first-derivative and

    • the diagram on-the-right showing the [ Natural-LOG Function ] itself .

  

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Defining [ e = 2.7182 8182 8.... ] :

• Now , let us pose this question :
  • What is the specific-and-precise value of [ x ] ,
    • when the [ Natural Logarithm of (x) ] is equal to [ 1.000000 ] ?

  And we were able to establish that :

  • this specific value of (x) is indeed :
    • [ 2.7182 8182 8.... ]

  And this specific value is , in general , denoted in mathematics by the symbol [ e ] , ( a slanted [ e ] )

  • i.e. :
    • [ e = 2.7182 8182 8 .... ]

• Thus , for [ x = 2.718282 ( = e ) ] , we then have this set of 2 diagrams below with :
  • the diagram on-the-left showing the first-derivative and

  • the diagram on-the-right showing the [ Natural-LOG Function ] itself .

  

  And the governing-and-defining equation for [ e ] is , of course :

  • 

• This then prepares us to move-on further :
  • to look at the Flexibility inherent within the Logarithmic Class of Functions , next .

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go to the next section : Section V - Flexibility inherent within the Logarithmic Class of Functions

go to the last section : Section III - Deriving the Logarithmic Functions

return to the MATH Functions Basics HomePage

Original dated 2014-12-07