We define the function by requiring that
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The value of this function is everywhere equal to its slope. Differentiating the first defining equation repeatedly we find that
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The second defining equation now tells us that for all The result is a particularly simple Taylor series:
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Let us check that a well-behaved function satisfies the equation
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if and only if
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We will do this by expanding the 's in powers of and and compare coefficents. We have
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and using the binomial expansion
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we also have that
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Voilà.
The function obviously satisfies and hence
So does the function
Moreover, implies
We gather from this
- that the functions satisfying form a one-parameter family, the parameter being the real number and
- that the one-parameter family of functions satisfies , the parameter being the real number
But also defines a one-parameter family of functions that satisfies , the parameter being the positive number
Conclusion: for every real number there is a positive number (and vice versa) such that
One of the most important numbers is defined as the number for which that is: :
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The natural logarithm is defined as the inverse of so Show that
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Hint: differentiate