Can I Find the Logarithmic Expansions of Log[x]?

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Discussion Overview

The discussion focuses on finding the logarithmic expansions of the function Log[x], specifically exploring series representations and their applicability over certain ranges of x. The conversation includes theoretical aspects of series expansions, particularly Taylor series, and considerations regarding the behavior of the logarithmic function near specific points.

Discussion Character

  • Exploratory
  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • One participant inquires about the series expansion of Log[x], emphasizing the urgency of the request.
  • Another participant notes that while there are many series for Log[x], they typically cover a limited range of x.
  • A participant explains that the Taylor series expansion can be used to express functions in terms of powers of (x-a), provided all derivatives are defined at x=a.
  • It is mentioned that Log[x] cannot be expanded in terms of powers of x due to the undefined nature of its derivatives at x=0.
  • A later reply reiterates that Log[x] cannot be expanded about zero in a series of nonnegative powers.
  • An example of a series expansion for log(1+x) is provided, applicable for |x|<1, showing a specific form of logarithmic expansion.

Areas of Agreement / Disagreement

Participants express some agreement regarding the limitations of expanding Log[x] around zero, but there are differing perspectives on the applicability and range of other series expansions for logarithmic functions.

Contextual Notes

There are limitations regarding the assumptions made about the range of x for which the series expansions are valid, particularly the undefined derivatives at x=0 and the specific conditions under which certain series can be applied.

Who May Find This Useful

This discussion may be useful for students and researchers interested in mathematical series, particularly those studying logarithmic functions and their expansions in various contexts.

mathelord
How Do I Find The Logarithmic Expansions Of Log[x],i Mean The Series Of Log[x].it Is Urgent
 
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The series expansion of any function can be obtained by Taylor's series expansion:
f(x)=f(a)+(x-a)f'(a)+(x-a)^2f"(a)/2!+(x-a)^3f"'(a)/3!+...

Using the above formula, any function can be expanded in terms of powers of (x-a), provided that all derivatives of f(x) are defined at x=a.

Note: logx can not be expanded in terms of powers of x, because the derivatives of logx are not defined at x=0.
 
mustafa said:
Note: logx can not be expanded in terms of powers of x, because the derivatives of logx are not defined at x=0.
I think you mean log x cannot be expanded about zero in a series of nonnegative powers.
 
example can be done via the log(1+x) series |x|<1

x-x^2/2+x^3/3...
 

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