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EDIT: I know b^a = e^(aln(b)), however, this uses exponentiation to define exponentiation.
So in school we're taught that exponentiation is repeated multiplication. However, this definition quickly falls apart when you have something like 2^pi. Afterall, what does it even mean to multiply 2 by itself pi times?
That definition gets even more wonky when you have things like (-2)^pi which isn't a real number.
What is the mathematical definition of exponentiation a^b that applies to all fields (real and complex) for ANY a or b?
My chief understanding of the exponential and the logarithm come from Spivak's wonderful book Calculus. He devotes a chapter to the definitions of both.
Think of exponentiation as some abstract operation 
(
is just some index, but you'll see why it's there) that takes a natural number 
and spits out a new number 
. You should think of 
.
To match our usual notion of exponentiation, we want it to satisfy a few rules, most importantly 
. Like how 
.
Now, we can extend this operation to the negative integers using this rule: take 
to be 
. then 
, like how 
.
Then we can extend the operation to the rational numbers, by taking 
. Like how 
.
Now, from here we can look to extend 
to the real numbers. This takes more work than what's happened up to now. The idea is that we want 
to satisfy the basic property of exponentiation: 
. This way we know it agrees with usual exponentiation for natural numbers, integers, and rational numbers. But there are a million ways to extend 
while preserving this property, so how do we choose?
Answer: Require 
to be continuous.
This way, we also have a way to evaluate 
for any real number 
: take a sequence of rational numbers 
converging to 
, then 
is 
. This seems like a pretty reasonable property to require!
Now, actually constructing a function that does this is hard. It turns out it's easier to define its inverse function, the logarithm 
, which is the area under the curve 
from 
to 
for 
. Once you've defined the logarithm, you can define its inverse 
. You can then prove that it has all the properties of the exponential that we wanted, namely continuity and 
. From here you can change the base of the exponential: 
.
To conclude: the real exponential function 
is defined (in fact uniquely) to be a continuous function 
satisfying the identity 
for all real 
and 
. One way to interpret it for real numbers is as a limit of exponentiating by rational approximations. Its inverse, the logarithm, can similarly be justified.
Finally, de Moivre's formula 
is what happens when you take the Taylor series expansion of 
and formally use it as its definition in the complex plane. This is more removed from intuition; it's really a bit of formal mathematical symbol-pushing.
or
(or any other irrational power, really). What does this mean?
$$a^\pi=a^{3.1415\ldots}=a^{3\ +\ 0.1\ +\ 0.04\ +\ 0.001\ +\ 0.0005\ +\ \cdots}=a^3\cdot a^{0.1}\cdot a^{0.04}\cdot a^{0.001}\cdot a^{0.0005}\cdots
a^\sqrt2=a^{1.4142\ldots}=a^{1\ +\ 0.4\ +\ 0.01\ +\ 0.004\ +\ 0.0002\ +\ \cdots}=a^1\cdot a^{0.4}\cdot a^{0.01}\cdot a^{0.004}\cdot a^{0.0002}\cdots$$
It is obvious that the general factor of this infinite product tends towards 
. Convergence then follows from the fact that each single decimal digit is in between 
and 
, meaning that 
is in between 
, and 
, where 
is the number of digits of 
.
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We could use the power series for the exponential function. In particular, we have $$ e^x = \sum_{k=0}^\infty \frac{x^k}{k!} $$ It would therefore follow that (for $a > 0$), $$ a^{x} = e^{\ln(a)x} = \sum_{k=0}^\infty \frac{[\ln(a)]^k x^k}{k!} $$ For other values of $a$, the value $a^x$ is not guaranteed to "make sense" in the context of the real numbers.
Exponentiation in general follows the following rules:
For all nonzero $x$
- $x=x^1$
- $x^a\cdot x^b = x^{a+b}$
- $x^0 = 1$
From these basic properties, we can see several things:
- $x^{-1}\cdot x = x^{-1}\cdot x^1 = x^0 = 1$ so $x^{-1}$ is in fact the multiplicative inverse of $x$
- $x^n = x\cdot x^{n-1}$
- $x^n = \underbrace{x\cdot x\cdots x}_{n~x\text{'s}}$ for a natural number $n$
- $x^{-1}=\frac{1}{x}$ since multiplicative inverses are unique.
- $x^{-a} = \frac{1}{x^a}$
- $(x^a)^b = x^{a\cdot b}$
For defining the exponentiation function for $x^n$ where $n$ is an integer, we can use the third result I listed above to relate it to multiplication and the fifth result if it were negative to simplify it first. For using exponents that are not integers, it requires a more careful definition, such as what Omnomnomnom provided in his answer.
A short example that uses several of these properties:
Simplify the following expression: $$\frac{2^3\cdot 3^1\cdot 2^1}{2^{-1}\cdot 3^1}$$
Answer: $2^{5}=32$