Final answer:
The value of log a(xz²/y⁻²) is calculated using properties of logarithms. Given log a(x) = 2, we can find log a(xz²/y⁻²) = 2 + 2log a(z) + 2log a(y). The actual values of log a(z) and log a(y) are needed for a numerical answer.
Step-by-step explanation:
The value of log a(xz²/y⁻²) can be calculated using the properties of logarithms and exponents. Given that log a(x) = 2, we can use the property that the logarithm of a product of two numbers is the sum of the logarithms of the two numbers and the logarithm of the number resulting from the division of two numbers is the difference between the logarithms of the two numbers.
Firstly, the logarithm of a product is equal to the sum of the logarithms, so we have log a(xz²) = log a(x) + log a(z²). Next, the logarithm of a quotient is equal to the difference of the logarithms, which gives us log a(xz²/y⁻²) = log a(x) + log a(z²) - log a(y⁻²).
The exponents can be factored out in front of the logarithms, resulting in log a(xz²/y⁻²) = log a(x) + 2log a(z) - (-2)log a(y). Finally, substituting the given value of log a(x), we have log a(xz²/y⁻²) = 2 + 2log a(z) + 2log a(y) which is the simplified form of the expression. To get the numerical value, values of log a(z) and log a(y) would be required.