How is irrational exponent defined? I am trying to understand the most significant jewel in mathematics - the Euler's formula. But first I try to re-catch my understanding of exponent function.
At the very beginning, exponent is used as a shorthand notion of multiplying several identical number together. For example, $5*5*5$ is noted as $5^3$. In this context, the exponent can only be $N$.
Then the exponent extends naturally to $0$, negative number, and fractions. These are easy to understand with just a little bit of reasoning. Thus the exponent extends to $Q$
Then it came to irrational number. I don't quite understand what an irrational exponent means? For example, how do we calculate the $5^{\sqrt{2}}$? Do we first get an approximate value of $\sqrt{2}$, say $1.414$. Then convert it to $\frac{1414}{1000}$. And then multiply 5 for 1414 times and then get the $1000^{th}$ root of the result? 
ADD 1
Thanks to the replies so far. 
In the thread recommended by several comments, a function definition is mentioned as below:
$$
ln(x) = \int_1^x \frac{1}{t}\,\mathrm{d}t
$$
And its inverse function is intentionally written like this:
$$
exp(x)
$$
And it implies this is the logarithms function because it abides by the laws of logarithms. 
I guess by the laws of logarithms that thread means something like this:
$$
f(x_1*x_2)=f(x_1)+f(x_2)
$$
But that doesn't necessarily mean the function $f$ is the logarithms function. I can think of several function definitions satisfying the above law.
So what if we don't explicitly name the function as $ln(x)$ but write it like this:
$$
g(x) = \int_1^x \frac{1}{t}\,\mathrm{d}t
$$
And its reverse as this:
$$
g^{-1}(x)
$$
How can we tell they are still the logarithm/exponent function as we know them? 
 A: Yes, you can approximate the result by approximating the irrational exponent with a rational number and proceed with computing integer powers and integer roots. But this does not give you much insight into what an irrational exponent might mean, and I think this is what you mostly care about. 
The best insightful explanation I've seen comes from Khalid at BetterExplained.com.
The short summary is that we have to stop seeing exponents as repeated multiplication and start seeing them as continuous growth functions, where $e$ plays a central role. 
So $5^{\sqrt2}$ can be written as $(e^{ln(5)})^\sqrt2 = e^{\sqrt2\cdot ln(5)}$. This can be interpreted as continuous growth for $1$ unit of time at a rate of $\sqrt2\cdot ln(5)$, or continuous growth for $\sqrt2$ units of time at a rate of $ln(5)$, or continuous growth for $ln(5)$ units of time at a rate of $\sqrt2$. They are all equivalent.
Check out these links for a much more detailed explanation:


*

*An Intuitive Guide To Exponential Functions & e

*How To Think With Exponents And Logarithms
A: One way of defining the real numbers is as equivalence classes of "the collection of all Cauchy sequences (or, equivalently, the collection of all increasing sequences with upper bound) of rational numbers" with "$\{a_n\}$ equivalent to $\{b_n\}$ if and only if $\{a_n- b_n\}$ converges to 0.  The essentially says that, for example, $\pi$ is "represented" by the infinite decimal 3.1415926....  From that definition, if a is an irrational number then there exist a sequence of rational numbers $r_1, r_2, r_3, ...$ that converges to a.  We then define $x^a$ to be the limit of the sequence $x^{r_1}, x^{r_2}, x^{r_3}, ...$.  Using the same example as before, $2^\pi$ is defined as the limit of the sequence $2^3, 2^{3.1}, 2^{3.14}, 2^{3.141}, 2^{3.1415}, 2^{3.14159}, 2^{3.141592}, 2^{31415926}, ...$. 
A: My two cents:
The definition has to do with the fact that $\mathbb R$ is by definition the completion of $\mathbb Q$:
$5^\sqrt{2}=\sup\{5^a:a\in \mathbb Q \wedge a<\sqrt 2\}$
You may show that the set on the RHS is bounded and nonempty, so $\sup$ exists.
So your approach is right (calculate $5^a$ for successive approximations of $\sqrt 2$ like 1.4, 1.41 and so on, and it will converge to the result).
I've seen the theory developed the other way around (first study the series, then define elementary functions as series), but this approach seems very "unnatural" to me, as it is not how this functions were developed historically.
A: The exponential function is an order-preserving bijection over the rationals. Filling in the holes gives an exponential function over the reals that is an order-preserving bijection. This is done by letting a^i be the supremum of {a^(p/q) | (p/q) 
