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Select the answer which is equivalent to the given expression using your calculator.

log_(729)243

(7)/(6)

(6)/(7)

(5)/(6)

(6)/(5)

Select the answer which is equivalent to the given expression using your calculator. $\newline$ $\log _{729} 243$ $\newline$ $\frac{7}{6}$ $\newline$ $\frac{6}{7}$ $\newline$ $\frac{5}{6}$ $\newline$ $\frac{6}{5}$

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Multiplication with rational exponents

Full solution

Q. Select the answer which is equivalent to the given expression using your calculator.

\newline

\log _{729} 243

\newline

\frac{7}{6}

\newline

\frac{6}{7}

\newline

\frac{5}{6}

\newline

\frac{6}{5}

Understand the problem: Understand the problem. $\newline$ We need to find the value of the logarithm $\log_{729}243$ . This means we are looking for the exponent that $729$ must be raised to in order to get $243$ .
Convert to exponential form: Convert the logarithm to an equivalent exponential form. $\newline$ The logarithmic equation $\log_{729}243 = x$ can be rewritten in exponential form as $729^x = 243$ .
Recognize number relationship: Recognize the relationship between the numbers $729$ and $243$ . Both $729$ and $243$ are powers of $3$ . Specifically, $729 = 3^6$ and $243 = 3^5$ .
Substitute powers of $3$ : Substitute the powers of $3$ into the exponential equation. $\newline$ $(3^6)^x = 3^5$
Apply power rule: Apply the power of a power rule. $\newline$ $3^{6x} = 3^5$
Solve for $x$ : Since the bases are the same, the exponents must be equal. $6x = 5$
Match solution to choices: Solve for $x$ . $x = \frac{5}{6}$
Match solution to choices: Solve for $x$ . $x = \frac{5}{6}$ Match the solution to the given answer choices. The value of $x$ is $\frac{5}{6}$ , which matches one of the given answer choices.

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