Question

Find the number of digits in the numeral for $(875{)}^{16}$

Answer 1

$875=25\times 35$

$=5^2\times 5\times 7$

$=5^3\times 7$

$(875{)}^{16}=(5^3\times 7{)}^{16}$

$=(5^3{)}^{16}\times 7^{16}$

$=5^{48}\times 7^{16}$

Taking $lo{g}_{10}$

$log(875{)}^{16}=lo{g}_{10}{5}^{48}+lo{g}_{10}{7}^{16}$

$\therefore log\left(ab\right)=lo{g}_{10}a+lo{g}_{10}b$

$=48lo{g}_{10}5+16lo{g}_{10}7$

$=48lo{g}_{10}\left(\frac{10}{2}\right)+16lo{g}_{10}7$

$(\because lo{g}_{10}10=1)$

$=48[1-log2]+16log7$

$=16\left[3\right(1-0.3010)+0.845]$

$16\times [3\times 0.699+0.845]$

$=47.072$

$\simeq 48$ digits.