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Any information on the following formula for natural logarithm (I looked in wikipedia and Mathworld but didn't see it). It came from another equation I was working on a bit ago, and I was curious about it as I didn't recall seeing it before (which doesn't mean I haven't), although it reminded me of some equations for

[tex] \ln{x} =\lim_{n\to\infty} \left[ \left (1- \frac{1}{x^{\frac{1}{n}}} \right) \times n \right ] [/tex]

For better visibility (bottom of the fraction is the nth root of x):

[tex]\ln{x} =\lim_{n\to\infty} \left[ \left (1- \frac{1}{ \sqrt[n]{x}} \right) \times n \right ] [/tex]

Or yet another form:

[tex] \ln{x} =\lim_{n\to\infty} \left[ \left (1- x^{- \frac{1}{n}} \right) \times n \right ] [/tex]

And I might have answered my own question with this last one... sheesh... anyways, still would like to read about it.

[tex] \ln{x} =\lim_{n\to\infty} \left[n- n\times x^{- \frac{1}{n}} \right ] [/tex]

Makes the derivative readily apparent, ehh? :D

*e*.[tex] \ln{x} =\lim_{n\to\infty} \left[ \left (1- \frac{1}{x^{\frac{1}{n}}} \right) \times n \right ] [/tex]

For better visibility (bottom of the fraction is the nth root of x):

[tex]\ln{x} =\lim_{n\to\infty} \left[ \left (1- \frac{1}{ \sqrt[n]{x}} \right) \times n \right ] [/tex]

Or yet another form:

[tex] \ln{x} =\lim_{n\to\infty} \left[ \left (1- x^{- \frac{1}{n}} \right) \times n \right ] [/tex]

And I might have answered my own question with this last one... sheesh... anyways, still would like to read about it.

[tex] \ln{x} =\lim_{n\to\infty} \left[n- n\times x^{- \frac{1}{n}} \right ] [/tex]

Makes the derivative readily apparent, ehh? :D

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