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log_(6)(1)/(36)=

$\log _{6} \frac{1}{36}=$

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Evaluate logarithms using properties

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Q.

\log _{6} \frac{1}{36}=

Recognizing the logarithm of a fraction: We are asked to find the value of the logarithm of $\frac{1}{36}$ with base $6$ . The first step is to recognize that the logarithm of a fraction can be expressed as the difference of the logarithms of the numerator and the denominator. $\newline$ $\log_{6}\left(\frac{1}{36}\right) = \log_{6}(1) - \log_{6}(36)$
Evaluating log base $6$ of $1$ : Now we need to evaluate $\log_{6}(1)$ and $\log_{6}(36)$ . The logarithm of any number to the same base is always $1$ , so $\log_{6}(1) = 0$ . This is because $6^{0} = 1$ .
Evaluating $\log_{6}(36)$ : Next, we need to evaluate $\log_{6}(36)$ . We know that $36$ is $6^2$ , so $\log_{6}(36) = \log_{6}(6^2)$ .
Simplifying $\log_{6}(6^{2})$ : Using the power property of logarithms, which states that $\log_{b}(a^{c}) = c \cdot \log_{b}(a)$ , we can simplify $\log_{6}(6^{2})$ to $2 \cdot \log_{6}(6)$ .
Simplifying $2 \times \log_{6} 6$ : Since $\log_{6}(6) = 1$ (because $6^{1} = 6$ ), we can further simplify $2 \times \log_{6}(6)$ to $2 \times 1 = 2$ .
Combining the results: Now we can combine our results to find the value of the original expression: $\log_{6}(\frac{1}{36}) = \log_{6}(1) - \log_{6}(36) = 0 - 2 = -2$ .

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