Does 9 appear as the leftmost digit in the decimal representation of any power of 2?

Does 7 appear as the leftmost digit in the decimal representation of any power of 37?

Is it possible to find a power of any counting number that has a given digit as its leftmost digit?

Also, is it possible to find a power of any counting number that begins with a given series of digits (e.g., is there a power of 24 that begins with 937)?

Prove that this is possible, or give a counter-example.

Bonus: What percentage of the powers of 2 have 1 as their leftmost digit?

Note: In finding the powers of "any counting number," exclude powers of ten, whose leftmost digit is always 1.

(In reply to Initial numbers by e.g.)

I'd say a part of the proof is missing... From the reference (at the PS) I'd say that taking the inequalities K+log(n) <= B.log(A) < K+log(n+1) in modulus 1 arithmetic, B.log(A) would satisfy this for infinitely many values of B.

Posted by Federico Kereki on 2004-02-16 09:53:45