Recognizing the logarithm of a fraction: We are asked to find the value of the logarithm of 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. log_(6)(1/36) = 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 base 6 of 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 base 6 of 6^2: Using the power property of logarithms, which states that log_b(a^c) = c * log_b(a), we can simplify log_(6)(6^2) to 2 * log_(6)(6).Simplifying 2 times log base 6 of 6: Since log_(6)(6) = 1 (because 6^1 = 6), we can further simplify 2 * log_(6)(6) to 2 * 1 = 2.Combining the results: Now we can combine our results to find the value of the original expression: log_(6)(1/36) = log_(6)(1) - log_(6)(36) = 0 - 2 = -2.

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

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Algebra 2

Evaluate logarithms using properties

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\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: $\log_{6}(\frac{1}{36}) = \log_{6}(1) - \log_{6}(36) = 0 - 2 = -2$ .