# First digit of powers of 2

Here are the first ten and second ten powers of two

 $2^{0}=1$ $2^{10}=1024$ $2^{1}=2$ $2^{11}=2048$ $2^{2}=4$ $2^{12}=4096$ $2^{3}=8$ $2^{13}=8192$ $2^{4}=16$ $2^{14}=16384$ $2^{5}=32$ $2^{15}=32768$ $2^{6}=64$ $2^{16}=65536$ $2^{7}=128$ $2^{17}=131072$ $2^{8}=256$ $2^{18}=262144$ $2^{9}=512$ $2^{19}=524288$

Notice the first digit of each power of two is the same as the power 10 larger.

Is it true that for any $n \ge 0$ $\large 2^{n} \text{ and } 2^{n+10}$ will begin with the same digit?

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