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Hint: Use logarithmic properties.

${a^{{{\log }_a}x}} = x$

$\log {m^n} = n\log m$

$\log x{\text{ is real }}\forall x > 0$

We have given the equation $2{\log _3}x + {\log _3}({x^2} - 3) = {\log _3}0.5 + {5^{{{\log }_5}({{\log }_3}8)}}$ which can be written as ${\log _3}{x^2} + {\log _3}({x^2} - 3) = {\log _3}0.5 + {\log _3}8$ and on further solving we’ll get $2{\log _3}x + {\log _3}({x^2} - 3) = {\log _3}0.5 + {\log _3}4$ . This equation is equivalent to the system

\[

\left\{ {\begin{array}{*{20}{c}}

{{x^2} > 0} \\

{{x^2} - 3 > 0} \\

{{x^2}({x^2} - 3) = 4}

\end{array}} \right. \\

\Rightarrow \left\{ {\begin{array}{*{20}{c}}

{x < 0{\text{ and }}x > 0{\text{ }}} \\

{x < \sqrt 3 {\text{ and x}} > \sqrt 3 } \\

{({x^2} - 4)({x^2} + 1) = 0}

\end{array}} \right. \\

\Rightarrow {x^2} - 4 = 0 \\

\therefore x = \pm 2,{\text{ but }}x > 0 \\

\]

Consequently, $x = 2$ is only the root of the given equation.

Note: When you are using log properties, be careful with the base. When the question says “$\ln $”, it means base is e. On the other hand, when it says “log”, it means the base Is 10, else wise questions will always write base.

${a^{{{\log }_a}x}} = x$

$\log {m^n} = n\log m$

$\log x{\text{ is real }}\forall x > 0$

We have given the equation $2{\log _3}x + {\log _3}({x^2} - 3) = {\log _3}0.5 + {5^{{{\log }_5}({{\log }_3}8)}}$ which can be written as ${\log _3}{x^2} + {\log _3}({x^2} - 3) = {\log _3}0.5 + {\log _3}8$ and on further solving we’ll get $2{\log _3}x + {\log _3}({x^2} - 3) = {\log _3}0.5 + {\log _3}4$ . This equation is equivalent to the system

\[

\left\{ {\begin{array}{*{20}{c}}

{{x^2} > 0} \\

{{x^2} - 3 > 0} \\

{{x^2}({x^2} - 3) = 4}

\end{array}} \right. \\

\Rightarrow \left\{ {\begin{array}{*{20}{c}}

{x < 0{\text{ and }}x > 0{\text{ }}} \\

{x < \sqrt 3 {\text{ and x}} > \sqrt 3 } \\

{({x^2} - 4)({x^2} + 1) = 0}

\end{array}} \right. \\

\Rightarrow {x^2} - 4 = 0 \\

\therefore x = \pm 2,{\text{ but }}x > 0 \\

\]

Consequently, $x = 2$ is only the root of the given equation.

Note: When you are using log properties, be careful with the base. When the question says “$\ln $”, it means base is e. On the other hand, when it says “log”, it means the base Is 10, else wise questions will always write base.