It's an extension of the usual laws of exponents for positive integers.

If we want some basic properties to be satisfied, then there is a way to extend this to negative integers, and zero. Thus, we can define integer powers of a positive number, like $10.$

Next come fractional indices, which are a bit subtle, but we may still define fractional powers of positive numbers in a way consistent with out extension to integer powers.

We can also extend to arbitrary real indices, but I think for your case it is sufficient to stop at rational indices.

So, you see, we can actually have such things as $10^{0.73578},$ and it is easy to see why such quantities cannot again be whole numbers. Basically the idea is that the function $10^x$ is monotonic. That is, it always increases as $x$ is increased. This is true for any positive base apart from $1,$ so there's nothing special about the $10.$ (The only caveat is that for bases less than $1,$ we have that $b^x$ decreases with increasing $x.$) Now, after having seen this, note that $10^0=1$ and $10^1=10.$ Thus, **if you pick any index $i$ between $0$ and $1$ (there are infinitely many of them), you shouldn't expect too much by way of an integral value for $10^i$ since there is a finite number of integers between $1$ and $10.$** In fact, most of such values are irrational. Your question of why some powers of $10$ can fail to be integral is thus answered -- the idea is that *that's what falls out of our extension of the idea of positive integral indices to arbitrary exponents, and it follows by the necessary fact that in any interval we always have infinitely many non-integers and finitely many integers. So it's a consequence of how we've chosen to define the function $10^x$ and its siblings, and the pigeonhole principle.*