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*?