Question

The value of ${\log }_{10}5\cdot {\log }_{10}20+({\log }_{10}2{)}^2$ is

Answer 1

${\log }_{10}5\cdot {\log }_{10}20+({\log }_{10}2{)}^2$
$={\log }_{10}\left(\frac{10}{2}\right)\cdot {\log }_{10}(10\times 2)+({\log }_{10}2{)}^2$
$=(1-{\log }_{10}2)(1+{\log }_{10}2)+({\log }_{10}2{)}^2$
$=1-({\log }_{10}2{)}^2+({\log }_{10}2{)}^2$
$=1$