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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