Natural Logarithms

An Important inequality:

Theorem: For $\QTR{Large}{0<x}$ , with equality if and only if

Proof:

For :

For :

so

and

For :

so

and

Corollary: , with equality if and only if

Proof:

Since we will use the notation for , we will just use that convention although obviously the Corollary is more general.

so $\ \hspace{1.5in}$

and

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An Alternative Definition of Natural Logarithms:

Theorem:

Proof: $\bigskip$

MATH

MATH

MATH

MATH

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An Alternative Definition:

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$\vspace{1pt}$ Theorem:

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$\vspace{1pt}$ Proof:

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Corollarys:

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Theorem:

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$\vspace{1pt}$ Proof:

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MATH

1/2+1/3+1/4+.......

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Definition:

is defined implicitly by the equation

And so: