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$Express the given expression as an integer or as a fraction in simplest form. log_(4)(4^(-(5)/(3))) Answer:$

Express the given expression as an integer or as a fraction in simplest form. $\newline$ $\log _{4}\left(4^{-\frac{5}{3}}\right)$ $\newline$ Answer:

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Multiplication with rational exponents

Full solution

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

\newline

\log _{4}\left(4^{-\frac{5}{3}}\right)

\newline

Answer:

Understand Logarithm Properties: Understand the properties of logarithms. The logarithm of a power of the base simplifies to the exponent times the logarithm of the base to the base itself. In this case, we have $\log_{4}(4^{x})$ . The property we use is $\log_{b}(b^{x}) = x$ , where $b$ is the base of the logarithm and $x$ is the exponent.
Apply Logarithm Power Rule: Apply the logarithm power rule. $\newline$ Using the property from Step $1$ , we can simplify $\log_{4}(4^{-\frac{5}{3}})$ to just the exponent, which is $-\frac{5}{3}$ , because the base of the logarithm and the base of the power are the same (both are $4$ ). $\newline$ $\log_{4}(4^{-\frac{5}{3}}) = -\frac{5}{3}$
Express Result as Fraction: Express the result as an integer or a fraction. $\newline$ The result from Step $2$ is already in the form of a fraction, which is the simplest form for this expression. There is no need to convert it to an integer because $-\frac{5}{3}$ cannot be expressed as an integer.

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