Why are the first few powers of $2^{10}$ a little more than those of 1000?
Solution 1:
Since $2^{10}=1024$: $$2^{10n}=(1000+24)^n=1000^n+24\cdot 1000^{n-1}n+...$$ Thus, as long as $24n$ remains a lot smaller than $1000$, then $2^{10n}$ will be near $1000^n$.
Solution 2:
A good "explanation" is that $\log_{10} 2 = 0.3010$.
Hence, $\log_{10} 2^{10} = 10 \log_{10} 2 = 3.01$, hence $2^{10}$ is very close to $10^3$.