How many digits will there be in 875^(16) ?


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Question Number 145359 by imjagoll last updated on 04/Jul/21

$How many digits will there be$ $in 875^{16} ?$
	Answered by Olaf_Thorendsen last updated on 04/Jul/21

	$If N is a n - digit number :$ $N = a_{n - 1} a_{n - 2} a_{n - 3} . . . . a_{2} a_{1} a_{0}$ $N = a_{n - 1}, a_{n - 2} a_{n - 3} . . . a_{3} a_{2} a_{1} \times 10^{n - 1}$ $\log_{10} N = \log_{10} (a_{n - 1}, a_{n - 2} . . . a_{3} a_{2} a_{1}) + n - 1$ $\Rightarrow n = ⌊ \log_{10} N ⌋ + 1$ $If N = 875^{16} :$ $n = ⌊ \log_{10} 875^{16} ⌋ + 1$ $n = ⌊ {16 \log}_{10} 875 ⌋ + 1$ $n = ⌊ 47, 72 . . . ⌋ + 1$ $n = 48$
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