Evaluate: log_(243)((1)/(9)) Answer:

Understand Problem and Identify Properties: Understand the problem and identify the properties of logarithms that can be used. We need to evaluate the logarithm of 1/9 with base 243. We can use the change of base formula for logarithms, which states that log base a of b can be written as log base c of b divided by log base c of a, where c is a new base we choose. In this case, we can choose base 3 because 243 and 9 are both powers of 3.Express Powers of 3: Express 243 and 9 as powers of 3. 243 is 3 raised to the 5th power (3^5), and 9 is 3 squared (3^2).Apply Change of Base Formula: Apply the change of base formula using base 3. log base 243 of 1/9 can be written as (log base 3 of 1/9) / (log base 3 of 243).Evaluate Logarithms Using Powers of 3: Evaluate the logarithms using the known powers of 3. log base 3 of 1/9 is the exponent we need to raise 3 to get 1/9, which is -2 because 3^(-2) = 1/9. log base 3 of 243 is the exponent we need to raise 3 to get 243, which is 5 because 3^5 = 243.Calculate Logarithm Value: Calculate the value of the logarithm. Now we have (log base 3 of 1/9) / (log base 3 of 243) which is (-2) / 5.Simplify Fraction: Simplify the fraction. -2 divided by 5 is -2/5.

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Evaluate: $\newline$ $\log _{243} \frac{1}{9}$ $\newline$ Answer:

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Precalculus

Quotient property of logarithms

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Q. Evaluate:

\newline

\log _{243} \frac{1}{9}

\newline

Answer:

Understand Problem and Identify Properties: Understand the problem and identify the properties of logarithms that can be used. $\newline$ We need to evaluate the logarithm of $\frac{1}{9}$ with base $243$ . We can use the change of base formula for logarithms, which states that $\log_{a}b$ can be written as $\frac{\log_{c}b}{\log_{c}a}$ , where $c$ is a new base we choose. In this case, we can choose base $3$ because $243$ and $9$ are both powers of $3$ .
Express Powers of $3$ : Express $243$ and $9$ as powers of $3$ . $243$ is $3$ raised to the $5$ th power ( $3^5$ ), and $9$ is $3$ squared ( $3^2$ ).
Apply Change of Base Formula: Apply the change of base formula using base $3$ . $\newline$ $\log_{243} \frac{1}{9}$ can be written as $\frac{\log_{3} \frac{1}{9}}{\log_{3} 243}$ .
Evaluate Logarithms Using Powers of $3$ : Evaluate the logarithms using the known powers of $3$ .
ext{log}_3 rac{1}{9} is the exponent we need to raise $3$ to get rac{1}{9}, which is $-2$ because 3^{-2} = rac{1}{9}.
$ext{log}_3 243$ is the exponent we need to raise $3$ to get $243$ , which is $3$ $0$ because $3$ $1$ .
Calculate Logarithm Value: Calculate the value of the logarithm. $\newline$ Now we have $(\log_{3} \frac{1}{9}) / (\log_{3} 243)$ which is $(-2) / 5$ .
Simplify Fraction: Simplify the fraction. $-2$ divided by $5$ is $-\frac{2}{5}$ .