Evaluate: log_(81)((1)/(243)) Answer:

Recognize Base and Argument: Recognize the base and the argument of the logarithm. We are given the logarithm log base 81 of 1/243. We need to evaluate this logarithm.Convert to Common Base: Convert the base and the argument into powers of a common base. Both 81 and 243 can be written as powers of 3, since 81 = 3^4 and 243 = 3^5.Rewrite Using New Expressions: Rewrite the logarithm using the new expressions for the base and the argument. log_(81)((1)/(243)) becomes log_(3^4)((1)/(3^5)).Apply Change of Base Formula: Apply the change of base formula for logarithms. The change of base formula is log_b(a) = log_c(a) / log_c(b), where c is a new base we choose. We can choose base 3 for convenience. log_(3^4)((1)/(3^5)) = log_3((1)/(3^5)) / log_3(3^4).Simplify Using Power Rule: Simplify the logarithms using the power rule. The power rule of logarithms states that log_b(a^n) = n * log_b(a). We apply this to both the numerator and the denominator. log_3((1)/(3^5)) / log_3(3^4) = (-5 * log_3(3)) / (4 * log_3(3)).Evaluate Logarithms: Evaluate the logarithms. Since log_3(3) = 1, we can simplify the expression further. (-5 * log_3(3)) / (4 * log_3(3)) = (-5 * 1) / (4 * 1) = -5 / 4.

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Evaluate: $\newline$ $\log _{81} \frac{1}{243}$ $\newline$ Answer:

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Precalculus

Quotient property of logarithms

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Q. Evaluate:

\newline

\log _{81} \frac{1}{243}

\newline

Answer:

Recognize Base and Argument: Recognize the base and the argument of the logarithm. We are given the logarithm $\log_{81} \frac{1}{243}$ . We need to evaluate this logarithm.
Convert to Common Base: Convert the base and the argument into powers of a common base. $\newline$ Both $81$ and $243$ can be written as powers of $3$ , since $81 = 3^4$ and $243 = 3^5$ .
Rewrite Using New Expressions: Rewrite the logarithm using the new expressions for the base and the argument. $\log_{81}\left(\frac{1}{243}\right)$ becomes $\log_{3^4}\left(\frac{1}{3^5}\right)$ .
Apply Change of Base Formula: Apply the change of base formula for logarithms. The change of base formula is $\log_b(a) = \frac{\log_c(a)}{\log_c(b)}$ , where $c$ is a new base we choose. We can choose base $3$ for convenience. $\log_{3^4}\left(\frac{1}{3^5}\right) = \frac{\log_3\left(\frac{1}{3^5}\right)}{\log_3(3^4)}$ .
Simplify Using Power Rule: Simplify the logarithms using the power rule. $\newline$ The power rule of logarithms states that $\log_b(a^n) = n \cdot \log_b(a)$ . We apply this to both the numerator and the denominator. $\newline$ $\log_3\left(\frac{1}{3^5}\right) / \log_3(3^4) = \frac{-5 \cdot \log_3(3)}{4 \cdot \log_3(3)}$ .
Evaluate Logarithms: Evaluate the logarithms. $\newline$ Since $\log_3(3) = 1$ , we can simplify the expression further. $\newline$ $(-5 \times \log_3(3)) / (4 \times \log_3(3)) = (-5 \times 1) / (4 \times 1) = -5 / 4$ .