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log_(27)(1)/(3)=

$\log _{27} \frac{1}{3}=$

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Evaluate logarithms using properties

Full solution

Q.

\log _{27} \frac{1}{3}=

Identify base and argument: Identify the base of the logarithm and the argument. $\newline$ The base of the logarithm is $27$ , and the argument is $\frac{1}{3}$ .
Recognize logarithm of $1$ : Recognize that any logarithm of $1$ is $0$ , regardless of the base. $\newline$ $\log_b(1) = 0$ for any base $b$ . $\newline$ Since the argument is $\frac{1}{3}$ , we need to find a way to relate $\frac{1}{3}$ to the base $27$ .
Express base as power of $3$ : Express the base $27$ as a power of $3$ . $\newline$ Since $27$ is a power of $3$ , we can write it as $27 = 3^3$ .
Rewrite logarithm with new base: Rewrite the logarithm with the new base expression. $\log_{27}(\frac{1}{3})$ can be rewritten as $\log_{3^3}(\frac{1}{3})$ .
Apply change of base formula: Apply the change of base formula for logarithms. $\newline$ The change of base formula is $\log_b(a) = \frac{\log_c(a)}{\log_c(b)}$ . $\newline$ Using this formula, we can rewrite the expression as $\frac{\log_{3}(\frac{1}{3})}{\log_{3}(3^3)}$ .
Evaluate logarithms: Evaluate the logarithms. $\newline$ $\log_{3}(\frac{1}{3})$ is asking ＂ $3$ to what power gives $\frac{1}{3}$ ?＂, which is $-1$ because $3^{-1} = \frac{1}{3}$ . $\newline$ $\log_{3}(3^3)$ is asking ＂ $3$ to what power gives $3^3$ ?＂, which is $3$ because $3^3 = 27$ . $\newline$ So we have $(-1) / 3$ .
Perform division for final answer: Perform the division to get the final answer. $\newline$ $(-1) / 3$ equals $-\frac{1}{3}$ .

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