An Approach to Mathematic Functions Basics

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

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Section VI - Expressing Logarithms via an Infinite Series

Summary for the Section :

• In this Section , we shall go thru the derivation process for :
  • expressing Logarithms via an Infinite Series .

  And the [ Infinite Series ] that we shall be developing here :

  • shall be primarily based on the Taylor Series Expansion Formula .

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Defining the particular Logarithmic Function for our use in this Section :

• Let the [ PHI-of-(x) ] be any Logarithmic Function such that :
  • its first-derivative at [ x = 1 ] is equal to [ K ] ,
    • with the value of [ K ] here being any non-zero value .

  And that-is-to-say , our [ PHI-of-(x) ] here is a Logarithmic Function

  • satisfying this particular condition :
    • 

• Since the [ PHI-of-(x) ] is a Logarithmic Class Function , it then follows immediately that :
  • the [ PHI-of-(x) ] possesses these 2 special properties :
    • First Property :
      • 

    • Second Property :
      • 

    ( as per our discussions on Logarithmic Functions in Section II of this paper . )

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The general format for the derivatives of the [ PHI-of-(x) ] :

• Let us now take a quick look at the derivatives of the [ PHI-of-(x) ] ,
  • and the first-derivative here is , of course :
    • 

  Consequently :

  • the 2nd-derivative is then :
    • 

  • the 3rd-derivative is then :
    • 

  • the 4th-derivative is then :
    • 

  • the 5th-derivative is then :
    • 

  • the 6th-derivative is then :
    • 

  • and so-on-and-so-forth .

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Evaluating the derivatives at ( x = 1 ) :

• Shortly below , we shall be doing a Taylor Series Expansion on the [ PHI-of-(x) ] ,
  • using [ x = 1 ] as the base-point for the expansion process .

  It is therefore appropriate for us to evaluate , at this point ,

  • the derivatives of the [ PHI-of-(x) ] at [ x = 1 ] ;
    • for our usage below .

• We then have , for the [ PHI-of-(x) ] at [ x = 1 ] :
  • the 1st-derivative being :
    • 

  • the 2nd-derivative being :
    • 

  • the 3rd-derivative being :
    • 

  • the 4th-derivative being :
    • 

  • the 5th-derivative being :
    • 

  • the 6th-derivative being :
    • 

  • and so-on-and-so-forth .

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Executing the Taylor Series Expansion :

• Let us now bring-in the general formula for a Taylor Series Expansion , i.e. the formula :
  • 

  We shall now proceed to do the expansion on the [ PHI-of-(x) ] via the above formula ,

  • using [ x = 1 ] as our base-point for the expansion process .

• As such , we now set up :
  • 

  And consequently , we have :

  • 

• Let us now also set up , for our ease-of-notation purposes ,
  • the dummy-variable [ delta ] , given by :
    • 

  Since we have already set up :

  • 

    the expression above for [ delta ] may now be re-written as follows :

    • 

  And consequently , we have the expression :

  • 

    for our usage immediately below .

• We are now ready to do the expansion and let us bring-in again the Taylor Series Expansion formula , i.e. :
  • 

  And for the substitutions into-the-formula to follow immediately , we shall be using these 4 items :

  • Item 1 :
    • 

  • Item 2 :
    • 

  • Item 3 :
    • the { derivatives for the [ PHI-of-(x) ] at [ x = 1 ] } from above ,

  • Item 4 :
    • 

• And upon full substitution, we do come up with this :
  • 

    

  And upon consolidation , we have :

  • 

  And the final expression for the [ infinite series ] here is then :

  • 

• And this [ infinite series ] immediately above , is then :
  • the core [ infinite series ] that we shall be using for the calculation of [ logarithms ] .
    • throughout the remainder of this paper .

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go to the next section : Section VII - The Logarithms of Natural Numbers

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

return to the MATH Functions Basics HomePage

Original dated 2014-12-07