Question

Find the value of
${\log }_{10}5.{\log }_{10}20+({\log }_{10}2{)}^2$

Answer 1

${\log }_{10}5{\log }_{10}20+({\log }_{10}2{)}^2$

${\log }_{10}5{\log }_{10}\left(4.5\right)+({\log }_{10}2{)}^2$

$\Rightarrow {\log }_{10}5[{\log }_{10}5+2\log 2]+({\log }_{10}2{)}^2$

$\Rightarrow ({\log }_{10}5{)}^2+2\log 2\log 5+({\log }_{10}2{)}^2$

$\Rightarrow ({\log }_{10}5+{\log }_{10}2{)}^2=(0.7+0.3{)}^2$

$=1$