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Evaluate:

log_(243)27
Answer:

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

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Evaluate logarithms

Full solution

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.

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