Question

Find the least integer $n$, such that $7^n>{10}^5,$ given that ${\log }_{10}343=\mathrm{2.5353.}$

Answer 1

${\log }_{10}343=\log 7^3=3\log 7=\mathrm{2.5353.}[\because \log a^m=m\log a]$
$\therefore \log 7=\frac{1}{3}\left(2.5353\right).Now{7}^n>{10}^5$ if
$n\log 7>5{\log }_{10}10$
or $n.\frac{1}{3}\left(2.5353\right)>5$
$\Rightarrow n>5.9164$
Therefore, least possible integer is $6$
Ans: 6