An Approach to Mathematic Functions Basics

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

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Section III - Deriving the Logarithmic Functions

Summary for the Section :

• In this Section , we shall go thru the derivation process for a special class of functions ,
  • namely :
    • the Logarithmic Functions .

  And we shall start-off the derivation process with :

  • the [ Governing Relation ] for the Logarithmic Class of Functions ,

    whereby :

    • the [ Governing Relation ] is the defining criteria that each-and-every Logarithmic Function must satisfy
      • in order to qualify as a Logarithmic Function .

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The [ Governing Relation ] for the Logarithmic Class of Functions :

• For each and every function [ PHI-of-(x) ] that qualifies as a Logarithmic Function ,
  • this particular relation :
    • 

      is always satisfied .

  And this then is the [ Governing Relation ] defining the Logarithmic Class of Functions .

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The First-and-Most-Important Property for the [ PHI-of-(x) ] :

• Let us now bring-in again the [ Governing Relation ] from above , i.e. :
  • 

  Let us now substitute the value of [ x = 1 ] into the above equation , and we have :

  • 

    yielding :

  • 

• And on cancelling terms , we do come up with this very special and specific relation :
  • 

  And this then is the [ First-and-Most-Important Property ] for the function [ PHI-of-(x) ] ,

  • namely that :
    • the value of the [ PHI of (1) ] is always equal to [ zero ] .

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The Search for Logarithms :

• Let us now take the first-derivative of the [ PHI-of-(x) ] with respect to [ x ] , and we have :
  • 

  And via a simple manipulation on the right-hand-side , we do come up with this alternate format for the same equation :

  • 

• Let us now apply the [ Governing Relation ] to this equation , and we have :
  • 

  And on cancelling terms , we have :

  • 

  Let us now re-write the above equation in this format below for our further manipulations :

  • 

    yielding :

  • 

• Let us now set up the the variable / quantity [ Q ] , defined via :
  • 

  We can then write :

  • 

  And we take note , at this point , that for [ x ] being [ not-equal-to-zero ] :

  • the value of [ Q ] also approaches [ zero ] as the value of [ h ] approaches [ zero ] ;

  and that-is-to-say :

  • 

  Thus , we can now re-write the above equation for the [ first-derivative of PHI ] in this manner :

  • 

• Let us now take a closer look at the term within-the-brackets on-the-right-hand-side of the above equation .

  And we bring-in , at this point , the first-derivative of the [ PHI-of-(x) ] evaluated at [ x = 1 ] , defined via :

  • 

  Let us again recall , from above , the [ First-and-Most-Important Property ] for the [ PHI-of-(x) ] ,

  • i.e. :
    • 

  And on substituting this into the equation immediately preceding , we have this :

  • 

• Let us now bring back the equation for the [ first-derivative of PHI ] we have previously developed above ,
  • i.e. this equation :
    • 

      for a quick comparison .

  And we see here that :

  • the term within-the-brackets on the right-hand-side of this equation
    • is in fact equivalent to
      • the { first-derivative of PHI evaluated at [ x = 1 ] } .

  Thus , we can now express the [ first-derivative of PHI ] in the following manner :

  • 

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Introducing the constant [ K ] :

• Let us now introduce the constant [ K ] , being the value :
  • 

  Two things to take special note of here :

  • [ K ] is the value of the { first-derivative of [ PHI-of-(x) ] evaluated at [ x = 1 ] } ,

  • the value of [ K ] is constant for any particular given function [ PHI-of-(x) ]
    • and the value of [ K ] does not vary with [ x ] .

• We can thus write the equation for the [ first-derivative of PHI ] as follows :
  • 

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Expressing the [ PHI-of-(x) ] as an integral :

• Now that we have an expression for the [ first-derivative of PHI ] in terms of [ x ] ,
  • we can simply integrate both sides of the equation to come up with an expression for the [ PHI-of-(x) ] .

  And on doing so , we have :

  • 

• Let us now try to set-up the limits for the Integrand on the right-hand-side .

  And again we recall , from above , the First-and-Most-Important Property for the [ PHI-of-(x) ] ,

  • i.e. :
    • 

  And what this tell us here then is that :

  • the limits for the Integrand should be set up as such so that :
    • the said Integrand does evaluate to [ zero ] when the corresponding target-value of [ x ] is equal [ 1 ] .

  Consequently , we can now write the equation for the [ PHI-of-(x) ] in this general format :

  • 

  Or alternately , we can write the same equation in this special-and-specific format below :

  • 

    for slightly better clarity .

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The Second-Most-Important Property for the [ PHI-of-(x) ] :

• Let us bring-in again , from above , the [ Governing Relation ] for the [ PHI-of-(x) ] ,
  • i.e. :
    • 

  Let us now substitute therein the values [ x = z ] and [ y = 1/z ] , where :

  • [ z ] is any real number other than [ zero ] .

  And on doing so , we have :

  • 

    yielding :

  • 

• Again let us recall the First-and-Most-Important Property for the [ PHI-of-(x) ] from above ,
  • i.e. :
    • 

  Applying this therein to the left-hand-side of the equation immediately preceding then yields us this :

  • 

  Consequently :

  • 

    and

    • this then is the Second-Most-Important Property for the [ PHI-of-(x) ] .

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Summarizing on the Logarithmic Class of Functions :

• Let us now summarize our findigs on the Logarithmic Class of Functions thus far :
  • IF :
    • the [ PHI-of-(x) ] is a Logarithmic Function satisfying the [ Governing Relation ] ,
      • i.e. the relation :
        • 

    THEN :

    • the [ PHI-of-(x) ] possesses these 2 properties :
      • 

        and

      • 

    • the general equation for the [ PHI-of-(x) ] is :
      • 

        where :

        • 

      and the [ first-derivative of PHI ] is given by :

      • 

• And with this general information now in-hand ,
  • we are ready to take a quick look at { Natural Logarithms } , next .

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go to the next section : Section IV - Defining Natural Logarithms

go to the last section : Section II - A Statement of Purpose for Part II

return to the MATH Functions Basics HomePage

Original dated 2014-12-07