Question
If $\log _{\sqrt{27}} x=2 \frac{2}{3},$ find $x$
✓
Answer
$\log _{\sqrt{27}} x=2 \frac{2}{3}$
$\therefore \log _{\sqrt{27}} x=\frac{8}{3}$
$\therefore x=(\sqrt{27})^{\frac{8}{3}} \ldots\left[\because \log _a x=b \Rightarrow x=a^b\right]$
$ \therefore x=\left(27^{\frac{1}{2}}\right)^{\frac{8}{3}}$
$ \therefore x=\left(3^{\frac{3}{2}}\right)^{\frac{8}{3}}$
$ \therefore x=3^{\left(\frac{3}{2}\right) \times\left(\frac{8}{3}\right)}$
$ \therefore x=3^4$
$ \therefore x=81$
$\therefore \log _{\sqrt{27}} x=\frac{8}{3}$
$\therefore x=(\sqrt{27})^{\frac{8}{3}} \ldots\left[\because \log _a x=b \Rightarrow x=a^b\right]$
$ \therefore x=\left(27^{\frac{1}{2}}\right)^{\frac{8}{3}}$
$ \therefore x=\left(3^{\frac{3}{2}}\right)^{\frac{8}{3}}$
$ \therefore x=3^{\left(\frac{3}{2}\right) \times\left(\frac{8}{3}\right)}$
$ \therefore x=3^4$
$ \therefore x=81$
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