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Evaluate:

log_(27)((1)/(243))
Answer:

Evaluate: $\newline$ $\log _{27} \frac{1}{243}$ $\newline$ Answer:

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Evaluate logarithms

Full solution

Q. Evaluate:

\newline

\log _{27} \frac{1}{243}

\newline

Answer:

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 $(1)/(243)$ . $\newline$ We need to express both the base and the argument as powers of a common base to simplify the logarithm.
Express as powers of $3$ : Express the base $27$ and the argument $\frac{1}{243}$ as powers of $3$ . $\newline$ $27$ is a power of $3$ because $27 = 3^3$ . $\newline$ $243$ is also a power of $3$ because $243 = 3^5$ . $\newline$ Therefore, $27$ $0$ can be written as $27$ $1$ since $\frac{1}{243}$ is the reciprocal of $243$ .
Rewrite using new expressions: Rewrite the logarithm using the new expressions for the base and the argument. $\log_{27}\left(\frac{1}{243}\right)$ becomes $\log_{3^3}(3^{-5})$ .
Apply logarithm power rule: Apply the logarithm power rule. $\newline$ The power rule of logarithms states that $\log_b(a^n) = n \cdot \log_b(a)$ . $\newline$ Using this rule, we can simplify $\log_{3^3}(3^{-5})$ to $-5 \cdot \log_{3^3}(3)$ .
Evaluate $\log_{3^3}(3)$ : Evaluate the logarithm $\log_{3^3}(3)$ . Since the base of the logarithm $(3^3)$ and the argument $(3)$ are powers of the same number, we can simplify this further. $\log_{3^3}(3)$ is asking ＂ $3$ to what power gives $3$ ?＂ The answer is $1$ because $3^1 = 3$ . Therefore, $\log_{3^3}(3) = 1$ .
Multiply by $-5$ : Multiply the result from Step $5$ by $-5$ . $\newline$ $-5 \times \log_{3^3}(3) = -5 \times 1 = -5$ . $\newline$ This is the value of the original logarithm.

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