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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