Digits of concatenated powers

Write out the full decimal expansions of $2^{1000}$ and $5^{1000}$ and concatenate them into a single number $n$ . How many digits does $n$ have?

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Digits of $m$ = $\lfloor\log_{10} m\rfloor+1$ . Note $\log_{10}2^{1000}+\log_{10}5^{1000}=1000$ .
For non-integer $x$ , $\lfloor x\rfloor+\lfloor 1000-x\rfloor=999$ ; add the two $+1$ 's.

Answer

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1001

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