TITLE: How is irrational exponent defined?
QUESTION [6 upvotes]: 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?

REPLY [3 votes]: 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 $2^3, 2^{3.1}, 2^{3.14}, 2^{3.141}, 2^{3.1415}, 2^{3.14159}, 2^{3.141592}, 2^{31415926}, ...$.