Final answer:
The expression log_3 (1/9)^3 simplifies to log_3 (1/729) when we apply the property of logarithms stating the logarithm of a number raised to an exponent equals the exponent times the logarithm of the base. This occurs because (1/9)^3 equals 1/729 which is 3^-6.
Step-by-step explanation:
To simplify the expression log_3 (1/9)^3, we first need to acknowledge the rules of logarithms and the properties of exponents. In this case, we apply the logarithmic property which states that the logarithm of a number raised to an exponent is the product of the exponent and the logarithm of the number. With this property, the expression can be rewritten as 3 × log_3 (1/9).
Furthermore, we know that 1/9 is 3^-2 because 9 is 3 squared, so we can substitute 3^-2 for 1/9 in the logarithm. The expression then simplifies to 3 × log_3 (3^-2), which can be simplified further by applying the logarithmic rule that log_b (b^x) = x. Here we have log_3 of 3 raised to the power of -2, which simplifies to -2. Multiplying this by 3 gives us -6.
Therefore, the final simplified expression is log_3 (1/729) because 3 to the power of -6 equals 1/729. Among the given options, log_3 (1/729) is the correct answer.