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

Identify base and argument: Identify the base of the logarithm and the argument. The base of the logarithm is 27, and the argument is (1)/(243). 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 1/243 as powers of 3. 27 is a power of 3 because 27 = 3^3. 243 is also a power of 3 because 243 = 3^5. Therefore, (1)/(243) can be written as 3^(-5) since 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)((1)/(243)) becomes log_(3^3)(3^(-5)).Apply logarithm power rule: Apply the logarithm power rule. The power rule of logarithms states that log_b(a^n) = n * log_b(a). Using this rule, we can simplify log_(3^3)(3^(-5)) to -5 * 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. -5 * log_(3^3)(3) = -5 * 1 = -5. This is the value of the original logarithm.

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

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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.