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 Solution by Charlie)

What's the connection to the title of the puzzle?
Edited on February 15, 2004, 1:01 pm

Posted by Penny on 2004-02-15 13:01:34