Answer
$1$
Work Step by Step
For $x\gt 0$ and $a$, a positive constant other than 1,
$\log_{a}x$ is the exponent to which $a$ must be raised in order to get $x$.
Thus, $\log_{a}x=m$ means $a^{m}=x$
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By definition, $\log_{81}3=x$ means
$x$ is the exponent with which we raise ($81$) to obtain $3.$
$3^{4}=81$
$\Rightarrow (3^{4})^{1/4}=81^{1/4}$
$\Rightarrow 3=81^{1/4}$ , so $\displaystyle \log_{81}3=\frac{1}{4}$
By definition, $\log_{3}81=y$ means
$y$ is the exponent with which we raise ($3$) to obtain $81.$
$3^{4}=81 \Rightarrow \log_{3}81=4$
Thus,
$\displaystyle \log_{81}3\cdot\log_{3}81=\frac{1}{4}\cdot 4=1$
