TITLE: What's the first digit of 2410^2410?
QUESTION [9 upvotes]: The first digit means the left most digit. 2410 is just an example and it can be replaced by any other numbers. 
Can any one help me to solve it?

REPLY [3 votes]: Using a calculatior we get that $\log_{10} (2410^{2410})=8150.66107260543188\ldots$.
Thus, $2410^{2410}$ has $8151$ digits, and the first one is $4$, since $10^{0.66107260543188}=4.582184853626742\ldots$.
In general, if we have a huge number $M$, then its first digit is the same as the first digit of $10^{\log_{10} M-\lfloor\log_{10} M\rfloor}$, where $\lfloor\cdot\rfloor$ is the integer part.
Incidentally, the second digit is 5, the third 8, the fourth 2 etc...