## The Exponential Property

We’ve defined the natural logarithm and shown that it is, in fact, a logarithm. That is, it’s a homomorphism from the multiplicative group of positive real numbers to the additive group of all real numbers. Now I assert that this function is in fact an isomorphism.

First off, the derivative of is , which is always positive for positive . Thus it’s always strictly increasing. That is, if then . So no two distinct numbers ever have the same natural logarithm, and the function is thus injective.

Flipping this around tells us that we definitely have some nonzero values for the function. For example, we know that . Now, since the real numbers are an Archimedean field, no matter how big a number we pick, there will be some natural number so that , where the latter inequality follows from the logarithmic property.

That is, no matter how large a number we pick, takes values at least that large. But because is continuous on a connected interval there must be some number with . Similarly, if then there will be some with , and thus . Thus the natural logarithm is surjective.

So, since our function is one-to-one and onto, it has an inverse function. We will call this function the “exponential” (denoted ), and define it to be the unique function satisfying

for all positive real and all real .

From here it’s straightforward to see that must be the inverse homomorphism. That is, given two real numbers and we know there must be (unique!) positive real numbers and with . Then we calculate

And it’s clear from here that . A homomorphism from the additive reals to the multiplicative positive reals like this is said to satisfy the “exponential property”, which is just the reverse of the logarithmic property from last time.

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Another parse error. :)

Comment by Jon | April 16, 2008 |

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Pingback by The Taylor Series of the Exponential Function « The Unapologetic Mathematician | October 7, 2008 |

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Pingback by The Exponential Series « The Unapologetic Mathematician | October 8, 2008 |

[...] Exponential Differential Equation So we long ago defined the exponential function to be the inverse of the logarithm, and we showed that it satisfied the exponential property. Now [...]

Pingback by The Exponential Differential Equation « The Unapologetic Mathematician | October 10, 2008 |

[...] The Circle Group Yesterday we saw that the unit-length complex numbers are all of the form , where measures the oriented angle from around to the point in question. Since the absolute value of a complex number is multiplicative, we know that the product of two unit-length complex numbers is again of unit length. We can also see this using the exponential property: [...]

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