If 2^P and 5^P start with the same (nonzero) 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 20061204 09:11:08 