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$Express the given expression as an integer or as a fraction in simplest form. log_(12)((1)/(root(4)(12))) Answer:$

Express the given expression as an integer or as a fraction in simplest form. $\newline$ $\log _{12}\left(\frac{1}{\sqrt[4]{12}}\right)$ $\newline$ Answer:

Full solution

Q. Express the given expression as an integer or as a fraction in simplest form.

\newline

\log _{12}\left(\frac{1}{\sqrt[4]{12}}\right)

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Answer:

Understand the expression: Understand the given expression. $\newline$ We need to find the value of the logarithm of the reciprocal of the fourth root of $12$ , with the base $12$ . $\newline$ $\log_{12}\left(\frac{1}{\sqrt[4]{12}}\right)$
Express as exponent: Express the fourth root of $12$ as an exponent. $\newline$ The fourth root of any number $x$ can be written as $x^{1/4}$ . $\newline$ $\sqrt[4]{12} = 12^{1/4}$
Rewrite using property: Rewrite the expression using the property of exponents. $\newline$ The reciprocal of $12^{1/4}$ can be written as $12^{-1/4}$ . $\newline$ $(1)/\left(\sqrt[4]{12}\right) = 12^{-1/4}$
Apply logarithm: Apply the logarithm. $\newline$ Now we have $\log_{12}(12^{-\frac{1}{4}})$ . $\newline$ According to the logarithm power rule, $\log_b(b^x) = x$ , where $b$ is the base of the logarithm. $\newline$ $\log_{12}(12^{-\frac{1}{4}}) = -\frac{1}{4}$
Verify result: Verify the result. $\newline$ Since the base of the logarithm and the base of the exponent are the same, the result is simply the exponent, which is $-\frac{1}{4}$ . This is an integer or a fraction in simplest form.