Question

Evaluate (i) ${\log }_{8}512$
(ii) ${\log }_{27}9$
(iii) ${\log }_{16}\left(\frac{1}{32}\right)$

Answer 1

(i) Let $x={\log }_{8}512$. Then
$8^x=512$ (exponential form)
$8^x=8^3⟹x=3$
$\therefore {\log }_{8}512=3$
(ii) Let $x={\log }_{27}9$. Then
${27}^x=9$ (exponential form)
${\left(3^3\right)}^x={\left(3\right)}^2$ (convert both sides to base three)
$3^{3x}=3^2⟹3x=2⟹x=\frac{2}{3}$
$\therefore {\log }_{27}9=\frac{2}{3}$
(iii) Let $x={\log }_{16}\left(\frac{1}{32}\right)$. Then
${16}^x=\frac{1}{32}$ (exponential form)
${\left(2^4\right)}^x=\frac{1}{{\left(2\right)}^5}$ (convert both sides to base two)
$2^{4x}=2^{-5}⟹4x=-5⟹x=-\frac{5}{4}$
$\therefore {\log }_{16}\left(\frac{1}{32}\right)=-\frac{5}{4}$