If 2^P and 5^P start with the same (non-zero) digit for positive integer P, what is that digit? Can you prove it must be so?

But this one is based on the log solution.

Consider 2^p to be expressed in scientific notation as a*10^m, and 5^p as b*10^n.  Their product, a*b*10^(m+n) must be a power of 10, 10^(m+n+1), so a*b = 10.

If a and b were exactly equal, they'd be sqrt(10), which begins with a 3.  If a were so low as exactly 3, b would be 3.333..., and when a got down to 2.999 (finite number of 9's, so just under 3), b would still start with 3 and there'd be a mismatch.  If a were as high as 3.99999, b would be2.5, a mismatch. Any further deviations would be further mismatches, so the only matching starting digit is 3.

Posted by Charlie on 2006-12-04 09:11:08