Evaluate: log_(243)27 Answer:

Identify base and number: Identify the base of the logarithm and the number whose logarithm is to be found. In log_(243)27, 243 is the base. Rewrite 27 as a power of 243. Since 243 is not a prime number, we need to express both 243 and 27 as powers of a common base, which is 3 in this case. 243 = 3^5 and 27 = 3^3.Rewrite as power: Rewrite the logarithm using the new expressions for 243 and 27. log_(243)27 becomes log_(3^5)(3^3).Use new expressions: Apply the change of base formula for logarithms. The change of base formula is log_b(a) = log_c(a) / log_c(b), where c is a new base. Using base 3, we get log_(3^5)(3^3) = log_3(3^3) / log_3(3^5).Apply change of base: Evaluate the logarithms. Since the base of the logarithms now matches the base of the exponents, we can simplify. log_3(3^3) = 3 because 3^3 = 27. log_3(3^5) = 5 because 3^5 = 243.Evaluate logarithms: Divide the results of the logarithms. log_(3^5)(3^3) = 3 / 5.Divide and simplify: Simplify the fraction if possible. The fraction 3 / 5 is already in its simplest form.

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

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Q. Evaluate:

\newline

\log _{243} 27

\newline

Answer:

Identify base and number: Identify the base of the logarithm and the number whose logarithm is to be found. $\newline$ In $\log_{243}27$ , $243$ is the base. $\newline$ Rewrite $27$ as a power of $243$ . $\newline$ Since $243$ is not a prime number, we need to express both $243$ and $27$ as powers of a common base, which is $3$ in this case. $\newline$ $243 = 3^5$ and $27 = 3^3$ .
Rewrite as power: Rewrite the logarithm using the new expressions for $243$ and $27$ . $\newline$ $\log_{243}27$ becomes $\log_{3^5}(3^3)$ .
Use new expressions: Apply the change of base formula for logarithms. $\newline$ The change of base formula is $\log_b(a) = \frac{\log_c(a)}{\log_c(b)}$ , where $c$ is a new base. $\newline$ Using base $3$ , we get $\log_{3^5}(3^3) = \frac{\log_3(3^3)}{\log_3(3^5)}$ .
Apply change of base: Evaluate the logarithms. $\newline$ Since the base of the logarithms now matches the base of the exponents, we can simplify. $\newline$ $\log_3(3^3) = 3$ because $3^3 = 27$ . $\newline$ $\log_3(3^5) = 5$ because $3^5 = 243$ .
Evaluate logarithms: Divide the results of the logarithms. $\log_{3^5}(3^3) = \frac{3}{5}$ .
Divide and simplify: Simplify the fraction if possible. $\newline$ The fraction $\frac{3}{5}$ is already in its simplest form.