# Chapter 12 - Exponential Functions and Logarithmic Functions - 12.3 Logarithmic Functions - 12.3 Exercise Set - Page 804: 110

$1$

#### 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 $a^{m}=x$ --- 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, $\displaystyle \log_{81}3\cdot\log_{3}81=\frac{1}{4}\cdot 4=1$

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