Evaluate: log_(27)((1)/(9)) Answer:

Understand logarithm and base: Understand the logarithm and its base. We are given the logarithm log base 27 of 1/9. We need to evaluate this expression.Express in terms of base: Express 1/9 in terms of the base 27. We know that 27 is 3^3 and 9 is 3^2. Therefore, 1/9 can be written as 3^(-2).Use change of base formula: Use the change of base formula. The change of base formula states that log base a of b = log base c of b / log base c of a. We can use the base of 3 for both the numerator and the denominator. log base 27 of 3^(-2) = log base 3 of 3^(-2) / log base 3 of 27Evaluate logarithms: Evaluate the logarithms. Since 27 is 3^3, log base 3 of 27 is 3. And since the base and the argument of the numerator are the same, log base 3 of 3^(-2) is -2. So, log base 27 of 3^(-2) = (-2) / 3Simplify expression: Simplify the expression. The expression (-2) / 3 simplifies to -2/3.

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Evaluate: $\newline$ $\log _{27} \frac{1}{9}$ $\newline$ Answer:

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Precalculus

Quotient property of logarithms

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Q. Evaluate:

\newline

\log _{27} \frac{1}{9}

\newline

Answer:

Understand logarithm and base: Understand the logarithm and its base. $\newline$ We are given the logarithm $\log_{27} \frac{1}{9}$ . We need to evaluate this expression.
Express in terms of base: Express $\frac{1}{9}$ in terms of the base $27$ . We know that $27$ is $3^3$ and $9$ is $3^2$ . Therefore, $\frac{1}{9}$ can be written as $3^{-2}$ .
Use change of base formula: Use the change of base formula. $\newline$ The change of base formula states that $\log_{a}b = \frac{\log_{c}b}{\log_{c}a}$ . We can use the base of $3$ for both the numerator and the denominator. $\newline$ $\log_{27}3^{-2} = \frac{\log_{3}3^{-2}}{\log_{3}27}$
Evaluate logarithms: Evaluate the logarithms. $\newline$ Since $27$ is $3^3$ , $\log_{3}{27}$ is $3$ . And since the base and the argument of the numerator are the same, $\log_{3}{3^{-2}}$ is $-2$ . $\newline$ So, $\log_{27}{3^{-2}} = \frac{-2}{3}$
Simplify expression: Simplify the expression. $\newline$ The expression $(-2) / 3$ simplifies to $-2/3$ .