Question
State, true or false : $\log _2 8=3$ and $\log _8=2=\frac{1}{3}$

Answer

Consider the equation
$
\log _2 8=3
$
$\Rightarrow 2^3=8$ $\ldots(1)$
Now consider the equation
$
\begin{aligned}
& \log _8 2=\frac{1}{3} \\
& \Rightarrow 8^{\frac{1}{3}}=2
\end{aligned}
$
$\Rightarrow\left(2^3\right)^{\frac{1}{3}}=2$ $\ldots(2)$
Both the equations (1) and (2) are correct.
Thus the given statements, $\log _2 8=3$ and $\log _8 2=\frac{1}{3}$ are true.

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