An Approach to Mathematic Functions Basics

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

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Section IV - Flexibility inherent within the Logarithmic Class of Functions

Summary for the Section :

• In this Section , we shall take a quick look at the flexibility inherent within the Logarithmic Class of Functions .

  Let us quickly recall that for any Logarithmic Function [ PHI-of-(x) ] , the equation for [ PHI ] is given by :

  • 

    where :

    • 

  In this Section , we shall attempt to answer this question :

  • What would happen if we were to let the value of [ K ] take on specific [ real values ]
    • varying from [ minus infinity ] to [ plus infinitity ] .

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Setting [ K = +2 / +1 / 0 / -1 / -2 ] :

• For our quick demonstration below , let us now :
  • set the value of [ K ] being equal to [ +2 ] , [ +1 ] , [ 0 ] , [ -1 ] and [ -2 ] respectively .

  Then :

  • For [ K = 2 ] and [ K = 1 ] , we then have this set of 2 diagrams below for the associated Logarithmic Curves :

    

  • For [ K = 0 ] , we then have this diagram below for the associated Logarithmic Curve :

    

    And we do take special note here that :

    • First-of-all :
      • The equation for the [ PHI-of-(x) ] here is :
        • 

    • Secondly :
      • This particular function does indeed qualify as a Logarithmic Function , because :
        • the [ Governing Relation ] for the Logarithmic Class of Functions ,
          • i.e. this relation :
            • 

            is always satisfied for the full-range-of-(x) from [ minus infinity ] to [ plus infinity ] .

    • Thirdly :
      • The first-derivative for the [ PHI-of-(x) ] here is :
        • 

        And this is consistent with :

        • 

          with the value of [ K ] being set-to-[ zero ] here for this particular [ PHI-of-(x) ] .

  • For [ K = -1 ] and [ K = -2 ] , we then have this set of 2 diagrams below for the associated Logarithmic Curves :

    

Consolidating the 5 diagrams :

• Let us now consolidate the 5 Logarithmic Curves we have above into a single diagram , as shown below :

  

  This diagram then tells us that :

  • When the value of [ K ] varies from [ minus infinity ] to [ plus infinity ] ,
    • there might be a problem at [ K = 0 ] .

• And because of this ,
  • we shall specifically leave out the case of [ K = 0 ] in our further discussions below .

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Setting up the general Logarithm Curve :

• Let us now set up a Logarithmic Curve ,
  • with the value of [ K ] being { any real-number other than [ zero ] } .

  We then bring-in this sample Logarithmic Curve with [ K = 1.5 ] ,

  • as per the diagram on-the-left below ,
    • for our demonstration purposes .

  

  Let us now mark-off a special-and-specific point on this Logarithmic Curve , namely :

  • the point with the [ X-Y co-ordinates ] being { e , PHI-of-(e) } ,

    i.e. :

    • the point [ Point B ] as shown in the diagram on-the-right above .

  Let us also mark-off 2 more points on the same diagram :

  • the point [ Point O ] ,
    • being the Point-Of-Origin with the [ X-Y co-ordinates ] being { 0 , 0 } ;

    and

  • the point [ Point A ] ,
    • being the point on the X-Axis with the [ X-Y co-ordinates ] being { e , 0 } .

• Thus ,
  • for the [ horizontal distance O-A ] , we have :
    • 

  • for the [ vertical distance A-B ] , we have :
    • 

      or ,

    • 

  Consequently :

  • the [ X-Y co-ordinates ] of [ Point B ] may now be expressed as { e , K } .

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The Tangent-Line at [ x = e ] :

• Let us now take a quick look at :
  • the line tangent-to-the-Logarithmic-Curve at [ Point B ] ,
    • as marked-off in magenta-color in the diagram on-the-left , below .

  

  And the slope of this [ tangent line ] is given by :

  • 

• Let us now focus our attention on [ Triangle O-A-B ] on the diagram on-the-right above .

  And we see here that :

  • 

  As such ,

  • the line [ Line O-B ] does have the same slope as the tangent-line passing thru [ Point B ] ;

    and

  • the 2 lines are therefore co-linear .

• We can now make this statement :
  • the { Line tangent-to-the-Logarithmic-Curve passing thru [ Point B ] } does pass thru the Point-Of-Origin [ Point O ] .

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At the Point-Of-Intersection of 2 Tangent-Lines :

• Let us now bring in a Logarithmic Curve for a given value of [ K ] ,
  • as shown below .

  

  Let us now mark-off the line tangent-to-curve and passing thru the Point-Of-Origin ,

  • as shown in the 2 diagrams below .

  

• We then have this set of 2 diagrams showing the Point-Of-Intersection of the 2 Tangent Lines .

  

  Of course ,

  • the value of [ 1.581977 ] is equal to :
    • [ e ] divided by { [ e ] minus [ 1 ] } .

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go to the next section : Section VI - Expressing Logarithms via an Infinite Series

go to the last section : Section IV - Natural Logarithms

return to the MATH Functions Basics HomePage

Original dated 2014-12-07