How-many-digits-will-there-be-in-875-16-

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} \dots . a_{2} a_{1} a_{0}

N = a_{n - 1}, a_{n - 2} a_{n - 3} \dots a_{3} a_{2} a_{1} \times 10^{n - 1}

\log_{10} N = \log_{10} (a_{n - 1}, a_{n - 2} \dots 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 \dots ⌋ + 1

n = 48