exponential function - 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?