Create a table of powers of 10 in binary starting with 101
then create a similar table in base 5 starting with 101
If you look at the lengths of the numbers in the two tables combined, prove there is exactly one each of length 2, 3, 4...
(In reply to further hint
I've started with compiling log(10,2)+1 and log(10,5)+1 to
see how the number of digit progresses, (4,7,10,14...etc-)
and got the idea, how it works....but that ,as you have mentioned,- is not a proof.
Anxious to see the formal proof re z+1/z for increasing powers of z.