TITLE: What are we doing when we raise $10$ to some decimal number?
QUESTION [7 upvotes]: This is something I am having some difficulty to understand.
When I have $10^x$ where $x$ is some integer  greater than $0$ , I understand  that this is equivalent to write $$\underbrace{10\cdot 10 \cdot 10 \cdots 10}_{x  \text {times}} $$ 
but when I have the case that $10$ is raised to some decimal number  $x$ I really lack the intuition of what's going on here.
I ask this question because ,for example ,when I read that $10^{0,48}=3,01...$ I simply don't understand how we have that $10$ raised to some number equals $3$.In general why is it that every number can be described as $10^x$ ? 
P.s I  guess it's important that I say that I am yet in high school,so you know what kind of answers might be appropriate for me.

REPLY [4 votes]: For the sake of illustration, I have made a plot of the powers of $2$ (those of $10$ grow too fast).

You see that they nicely align on a smooth and regular curve. Then it is a natural thing to define the powers for non-integer numbers too. And by continuity of the curve, for every number $y$, there will be some $x$ such that $y=2^x$.
How exactly we can compute these is another matter, but here is a hint: it is easy to show that $\sqrt{2^{2n}}=2^n$. Then, we can generalize to real $x$ by admitting $\sqrt{2^x}=2^{x/2}$. This tells us $2^{0.5}=\sqrt{2}, 2^{1.5}=2\sqrt2,2^{0.25}=\sqrt{\sqrt{2}},2^{0.75}=\sqrt{\sqrt{2}}\sqrt{2}\cdots$