Show that these two numbers have the same number of digits

Note the only way $2^n+1$ can have one more digit than $2^n$ is if $2^n$ ended in a $9$ (actually ends is $\cdots 999999$ but that is not important). $2^n$ can never end in a $9$.

In order for them to have a different number of digits, $2^n+1$ must be exactly a power of 10. But that's impossible, since $2^n+1$ is odd.