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Use the Change of Base Formula to evaluate
log_(5)92. Then convert
log_(5)92 to a logarithm in base 3. Round to the nearest thousandth.

$4$ . Use the Change of Base Formula to evaluate $\log _{5} 92$ . Then convert $\log _{5} 92$ to a logarithm in base $3$ . Round to the nearest thousandth.

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Properties of logarithms: mixed review

Full solution

Q.

4

. Use the Change of Base Formula to evaluate

\log _{5} 92

. Then convert

\log _{5} 92

to a logarithm in base

3

. Round to the nearest thousandth.

Apply Change of Base Formula: Use the Change of Base Formula: $\log_b(a) = \frac{\log_c(a)}{\log_c(b)}$ . Here, $a = 92$ and $b = 5$ . We'll use the common base of $10$ . $\log_{5}92 = \frac{\log(92)}{\log(5)}$ .
Calculate log values: Calculate $\log(92)$ and $\log(5)$ using a calculator. $\newline$ $\log(92) \approx 1.9638$ , $\log(5) \approx 0.6990$ .
Divide log values: Divide $\log(92)$ by $\log(5)$ to find $\log_{5}92$ . $\newline$ $\log_{5}92 \approx \frac{1.9638}{0.6990} \approx 2.8104$ .
Convert to base $3$ : Now convert $\log_{5}92$ to a logarithm in base $3$ using the Change of Base Formula again. $\newline$ $\log_{3}92 = \frac{\log_{5}92}{\log_{5}3}.$
Calculate $\log_{5}3$ : Calculate $\log_{5}3$ using a calculator. $\newline$ $\log_{5}3 = \frac{\log(3)}{\log(5)} \approx \frac{0.4771}{0.6990} \approx 0.6826$ .
Divide log values: Divide $\log_{5}92$ by $\log_{5}3$ to find $\log_{3}92$ . $\newline$ $\log_{3}92 \approx \frac{2.8104}{0.6826} \approx 4.116$ .

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