Evaluate logarithm with base 81: We need to evaluate the logarithm of 1/3 with base 81. The base of the logarithm is 81, which is 3^4.Express 1/3 as 3^-1: We can express 1/3 as 3^-1 since dividing by 3 is the same as multiplying by 3 to the power of -1.Rewrite logarithm with new expressions: Now we rewrite the logarithm using the new expressions for the base and the argument: log_(3^4)(3^-1).Apply change of base formula: We can apply the change of base formula for logarithms, which states that log_b(a) = log_c(a) / log_c(b). In this case, we can change the base to 3 because it is a common base for both 81 and 1/3. This gives us log_3(3^-1) / log_3(3^4).Simplify logarithmic expressions: Since the logarithm of a number to the same base is 1, log_3(3^-1) simplifies to -1 and log_3(3^4) simplifies to 4.Divide the results: Now we divide the two results: -1 / 4.Final answer: The final answer is -1/4. This completes the evaluation of the logarithm.

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$\log _{81} \frac{1}{3}=$

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Algebra 2

Evaluate logarithms using properties

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\log _{81} \frac{1}{3}=

Evaluate logarithm with base $81$ : We need to evaluate the logarithm of $\frac{1}{3}$ with base $81$ . The base of the logarithm is $81$ , which is $3^4$ .
Express $\frac{1}{3}$ as $3^{-1}$ : We can express $\frac{1}{3}$ as $3^{-1}$ since dividing by $3$ is the same as multiplying by $3$ to the power of $-1$ .
Rewrite logarithm with new expressions: Now we rewrite the logarithm using the new expressions for the base and the argument: $\log_{3^4}(3^{-1})$ .
Apply change of base formula: We can apply the change of base formula for logarithms, which states that $\log_b(a) = \frac{\log_c(a)}{\log_c(b)}$ . In this case, we can change the base to $3$ because it is a common base for both $81$ and $\frac{1}{3}$ . This gives us $\frac{\log_3(3^{-1})}{\log_3(3^4)}$ .
Simplify logarithmic expressions: Since the logarithm of a number to the same base is $1$ , $\log_3(3^{-1})$ simplifies to $-1$ and $\log_3(3^4)$ simplifies to $4$ .
Divide the results: Now we divide the two results: $-1 / 4$ .
Final answer: The final answer is $-\frac{1}{4}$ . This completes the evaluation of the logarithm.