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.)

And if X^Y begins with M, then there exists P and Q such that:
P x log Q <= X^Y <= P+M x log Q/M
(of course the logs are taken in base 10)

Further, if one plots this as a vector on the complex plane, one will quickly see that the angle theta is a monotonically increasing function, while the vector length oscillates with a frequency approaching e^π.

The proof that this is so is left as an exercise to the reader.

- SK

P.S. I didn't understand my post either...

Posted by SilverKnight on 2004-02-16 09:14:05