# Same first digit

We raise the numbers 2 and 5 to the same positive integer power and get two numbers that have the same first (leftmost) digit. What are the possible values of the first digit?

• Well... I believe the only possible number is 3. – Samuel Mar 28 '17 at 16:35
• First as in leftmost or rightmost? – John Mar 28 '17 at 16:37
• @John: for the rightmost the only answer is $1$ with the power being $0$. – Ross Millikan Mar 28 '17 at 16:42
• I don't know how to find the first digit... If it were possible to have the total number n of digits (length of number) and then divide by 10^(n-1) and then round down the quotient... – Samuel Mar 28 '17 at 16:44
• Just thinking out loud: We can also take the logarithm "L" of the number, take its integer part "i" and then divide by 10^i, then round down the quotient. I don't know if any of these things make sense! – Samuel Mar 28 '17 at 16:53

Hint: what is the product $2^n \cdot 5^n?$