Digits of 2 to the 1000 next to 5 to the 1000

Write the decimal forms of $2^{1000}$ and $5^{1000}$ side by side to make one long number. How many digits does it have?

$1001$. The digit counts are $\lfloor1000\log_{10}2\rfloor+1$ and $\lfloor1000\log_{10}5\rfloor+1$. The two logs sum to exactly $1000$ and neither is an integer, so their floors sum to $999$, giving $999+2=1001$.