Question

If $x={\left(0.15\right)}^{20}$, then find the mantissa of the logarithm of $x$ to the base $10$.

Assume ${\log }_{10}2=0.301$ and ${\log }_{3}10=0.477$.

Answer 1

$\log x=\log {\left(0.15\right)}^{20}$

$=20\log \left(\frac{15}{100}\right)$
$=20[\log 15-2]$
$=20[\log 3+\log 5-2]$
$=20[\log 3+1-\log 2-2]$ $(\because {\log }_{10}5={\log }_{10}(\frac{10}{2}\left)\right)$
$=20[-1+\log 3-\log 2]$
$=20[-1+0.477-0.301]$
$=-20\times 0.824=-16.48$

$=-16-0.48$

$=-16-1+1-0.48=-17+0.52$

$=\overline{17}.52$
Hence, Characteristic$=-17$ and mantissa$=0.52$
Ans: 0.52