Question

If ${\left(0.2\right)}^x=2$ and ${\log }_{10}2=0.3010$, then what is the value of $x$

Answer 1

$(0.2{)}^x=2$

Taking log on both sides,

$\log (0.2{)}^x=\log 2$

$x\log 0.2=0.3010\dots (n{\log }_{a}b={\log }_a{b}^n)$

$x(log2-log10)=0.3010\dots (\log \left(\frac{a}{b}\right)=\log a-\log b)$

$x(0.3010-1)=0.3010$

$x(-0.6990)=0.3010$

$x=\frac{-3010}{6990}=-0.43$

$x=-0.43$