Without the use of calculators or software establish which number is larger: $\newline$ $\log_{189}1323$ or $\log_{63}147$ .

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Precalculus

Product property of logarithms

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Q. Without the use of calculators or software establish which number is larger:

\newline

\log_{189}1323

\log_{63}147

Use Change of Base Formula: Use the change of base formula to compare the two logarithms. $\newline$ $\log_{189}1323 = \frac{\log(1323)}{\log(189)}$ $\newline$ $\log_{63}147 = \frac{\log(147)}{\log(63)}$
Simplify Bases and Numbers: Simplify the bases and the numbers inside the logarithms by finding common factors. $\newline$ $189 = 3^3 \times 7$ $\newline$ $63 = 3^2 \times 7$ $\newline$ $1323 = 3^3 \times 7^2$ $\newline$ $147 = 3 \times 7^2$
Cancel Common Factors: Simplify the expressions by canceling out common factors. $\newline$ $\log_{189} 1323 = \frac{\log(3^3 \cdot 7^2)}{\log(3^3 \cdot 7)}$ $\newline$ $\log_{63} 147 = \frac{\log(3 \cdot 7^2)}{\log(3^2 \cdot 7)}$
Apply Power Rule: Use the power rule of logarithms to bring down the exponents. $\newline$ $\log_{189}1323 = \frac{3 \cdot \log(7) + 2 \cdot \log(7)}{3 \cdot \log(3) + \log(7)}$ $\newline$ $\log_{63}147 = \frac{\log(3) + 2 \cdot \log(7)}{2 \cdot \log(3) + \log(7)}$
Combine Terms: Combine the terms with common logarithms. $\newline$ $\log_{189}1323 = \frac{5 \cdot \log(7)}{3 \cdot \log(3) + \log(7)}$ $\newline$ $\log_{63}147 = \frac{2 \cdot \log(7) + \log(3)}{2 \cdot \log(3) + \log(7)}$
Compare Numerators and Denominators: Compare the numerators and denominators of both expressions. $\newline$ For $\log_{189}1323$ , the numerator is $5 \times \log(7)$ and the denominator is $3 \times \log(3) + \log(7)$ . $\newline$ For $\log_{63}147$ , the numerator is $2 \times \log(7) + \log(3)$ and the denominator is $2 \times \log(3) + \log(7)$ .
Correct Comparison Mistake: Notice that the numerator of $\log_{189}1323$ is larger than the numerator of $\log_{63}147$ , since $5 \times \log(7) > 2 \times \log(7) + \log(3)$ . However, the denominator of $\log_{189}1323$ is also larger than the denominator of $\log_{63}147$ , since $3 \times \log(3) + \log(7) > 2 \times \log(3) + \log(7)$ .
Correct Comparison Mistake: Notice that the numerator of $\log_{189}1323$ is larger than the numerator of $\log_{63}147$ , since $5 \times \log(7) > 2 \times \log(7) + \log(3)$ . However, the denominator of $\log_{189}1323$ is also larger than the denominator of $\log_{63}147$ , since $3 \times \log(3) + \log(7) > 2 \times \log(3) + \log(7)$ . Since both the numerator and denominator of $\log_{189}1323$ are larger than those of $\log_{63}147$ , we cannot directly conclude which fraction is larger without further simplification or numerical evaluation.
Correct Comparison Mistake: Notice that the numerator of $\log_{189}1323$ is larger than the numerator of $\log_{63}147$ , since $5 \times \log(7) > 2 \times \log(7) + \log(3)$ . However, the denominator of $\log_{189}1323$ is also larger than the denominator of $\log_{63}147$ , since $3 \times \log(3) + \log(7) > 2 \times \log(3) + \log(7)$ . Since both the numerator and denominator of $\log_{189}1323$ are larger than those of $\log_{63}147$ , we cannot directly conclude which fraction is larger without further simplification or numerical evaluation. Realize that a mistake was made in the previous step. The comparison of the numerators and denominators was incorrect. We need to correct this and compare the values properly.