If $\log_27=x$ is $x$ irrational or rational? $x$ will definitely be between $2$ and $3$, but why does it have to irrational? Let’s assume that you don’t have the log table and no calculator, then how do you determine if it’s rational or irrational?

It’s probably an obvious answer like $x$ will always be irrational and all that, but I’d still like to know. 

Thanks a lot.
Hint $\log_2 (7) =x \Leftrightarrow 2^x=7$. Now, if $x=\frac{m}{n}$ would be rational then
$$2^m=7^n$$
Another approach:

$\log _2 a=x$ is integer if and only if $a=2^n; n∈N$

But $7=2^3-1$, so x is not integer which means it is a fraction or a rational number with denominator greater than unity.