Question

Find $x$ if
a) $2^x=22$
b) $3^x=243$

Answer 1

$\left(a\right)2^x=22$

$\Rightarrow {\log }_2{2}^x={\log }_{2}22$

$\Rightarrow x{\log }_{2}2={\log }_{2}2\times 11$ using $\log ab=\log a+\log b$

$\Rightarrow x{\log }_{2}2={\log }_{2}2+{\log }_{2}11$

$\Rightarrow x=1+{\log }_{2}11$ since ${\log }_{a}a=1$

$\left(b\right)3^x=243$

$\Rightarrow {\log }_3{3}^x={\log }_{3}243$

$\Rightarrow x{\log }_{3}3={\log }_3{3}^5$

$\Rightarrow x{\log }_{3}3=5{\log }_{3}3$

$\Rightarrow x=5$ since ${\log }_{a}a=1$