Evaluate: log_(9)((1)/(3)) Answer:

Identify base and argument: Identify the base and the argument of the logarithm. We are given the logarithm log_(9)((1)/(3)). The base is 9, and the argument is 1/3.Use change of base: Use the change of base formula to rewrite the logarithm in terms of common 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 10 for common logarithms. log_(9)((1)/(3)) = log((1)/(3)) / log(9)Evaluate common logarithms: Evaluate the common logarithms. We know that log(1/3) is the logarithm of 1/3 to the base 10, and log(9) is the logarithm of 9 to the base 10. We can simplify these because 9 is 3^2 and 1/3 is 3^(-1). log((1)/(3)) = log(3^(-1)) = -1 * log(3) log(9) = log(3^2) = 2 * log(3)Divide results: Divide the two results. Now we divide the two results from Step 3 to find the value of the original logarithm. log_(9)((1)/(3)) = (-1 * log(3)) / (2 * log(3))Simplify expression: Simplify the expression. Since log(3) appears in both the numerator and the denominator, they cancel each other out. log_(9)((1)/(3)) = -1 / 2

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Evaluate: $\newline$ $\log _{9} \frac{1}{3}$ $\newline$ Answer:

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Precalculus

Quotient property of logarithms

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Q. Evaluate:

\newline

\log _{9} \frac{1}{3}

\newline

Answer:

Identify base and argument: Identify the base and the argument of the logarithm. $\newline$ We are given the logarithm $\log_{9}\left(\frac{1}{3}\right)$ . The base is $9$ , and the argument is $\frac{1}{3}$ .
Use change of base: Use the change of base formula to rewrite the logarithm in terms of common logarithms. $\newline$ 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 $10$ for common logarithms. $\newline$ $\log_{9}\left(\frac{1}{3}\right) = \frac{\log\left(\frac{1}{3}\right)}{\log(9)}$
Evaluate common logarithms: Evaluate the common logarithms. $\newline$ We know that $\log(\frac{1}{3})$ is the logarithm of $\frac{1}{3}$ to the base $10$ , and $\log(9)$ is the logarithm of $9$ to the base $10$ . We can simplify these because $9$ is $3^2$ and $\frac{1}{3}$ is $3^{-1}$ . $\newline$ $\frac{1}{3}$ $0$ $\newline$ $\frac{1}{3}$ $1$
Divide results: Divide the two results. $\newline$ Now we divide the two results from Step $3$ to find the value of the original logarithm. $\newline$ $\log_{9}\left(\frac{1}{3}\right) = \frac{-1 \times \log(3)}{2 \times \log(3)}$
Simplify expression: Simplify the expression. $\newline$ Since $\log(3)$ appears in both the numerator and the denominator, they cancel each other out. $\newline$ $\log_{9}\left(\frac{1}{3}\right) = -\frac{1}{2}$