First digit of powers of 2

Here are the first ten and second ten powers of two

20=12^{0}=1210=10242^{10}=1024
21=22^{1}=2211=20482^{11}=2048
22=42^{2}=4212=40962^{12}=4096
23=82^{3}=8213=81922^{13}=8192
24=162^{4}=16214=163842^{14}=16384
25=322^{5}=32215=327682^{15}=32768
26=642^{6}=64216=655362^{16}=65536
27=1282^{7}=128217=1310722^{17}=131072
28=2562^{8}=256218=2621442^{18}=262144
29=5122^{9}=512219=5242882^{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 n0n \ge 0 2n and 2n+10\large 2^{n} \text{ and } 2^{n+10} will begin with the same digit?

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