The Unapologetic Mathematician

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 $\ln(x)$ is $\frac{1}{x}$, which is always positive for positive $x$. Thus it’s always strictly increasing. That is, if $0<x<y$ then $\ln(x)<\ln(y)$. 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 $0<\ln(2)$. Now, since the real numbers are an Archimedean field, no matter how big a number $y>0$ we pick, there will be some natural number $n$ so that $y<n\ln(2)=\ln(2^n)$, where the latter inequality follows from the logarithmic property.

That is, no matter how large a number we pick, $\ln$ takes values at least that large. But because $\ln$ is continuous on a connected interval there must be some number $x$ with $\ln(x)=y$. Similarly, if $y<0$ then there will be some $x$ with $\ln(x)=-y$, and thus $\ln(\frac{1}{x})=y$. 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 $\exp$), and define it to be the unique function satisfying

$\exp( \ln(x) )=x$
$\ln(\exp(y))=y$

for all positive real $x$ and all real $y$.

From here it’s straightforward to see that $\exp$ must be the inverse homomorphism. That is, given two real numbers $y_1$ and $y_2$ we know there must be (unique!) positive real numbers $x_1$ and $x_2$ with $\ln(x_i)=y_i$. Then we calculate

$\exp(y_1+y_2)=\exp(\ln(x_1)+\ln(x_2))=\exp(\ln(x_1 x_2))=$
$x_1x_2=\exp(\ln(x_1))\exp(\ln(x_2))=\exp(y_1)\exp(y_2)$

And it’s clear from here that $\exp(0)=1$. 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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Posted by John Armstrong | Analysis, Calculus

8 Comments	»

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Pingback by | April 10, 2008 | Reply

2. Another parse error.

Comment by | April 16, 2008 | Reply

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4. [...] run from zero all the way up to infinity. Now the same sort of argument as we used to construct the exponential function gives us an inverse sending any nonnegative number to a unique nonnegative square [...]

Pingback by | August 19, 2008 | Reply

5. [...] one that’s particularly nice is the exponential function . We know that this function is its own derivative, and so it has infinitely many derivatives. In [...]

Pingback by | October 7, 2008 | Reply

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Pingback by | October 8, 2008 | Reply

7. [...] 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 [...]

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8. [...] 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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