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.

If A^B starts with N, then there exists K so that Nx10^K<=A^B<(N+1)x10^K. Taking decimal logarithms, K+log(N)<=Bxlog(A)
PS. The Kronecker theorem can be found at
http://mathworld.wolfram.com/KroneckersApproximationTheorem.html

Posted by e.g. on 2004-02-16 08:07:16