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.
All of these questions are simple when thinking in terms of a slide rule. For any number, counting or otherwise, if the log base 10 is irrational, then the set of all of its powers completely cover the slide rule, with equal density.
That makes the answers as follows.
a) Yes. the percentage of powers of 2 that begin with a 9 = log(10) - log(9) = 1 - log(9)
b) Yes. The percentage of powers of 37 that start with 7 = log(8) - log(7)
c) Yes, unless the counting number is a power of 10, in which case its logarithm is rational. All powers of a power of 10 begin with a 1.
Bonus) log(2), approximately 30.1%
Edited on July 16, 2012, 2:59 pm
Slide Rule logic (spoiler)
