Final answer:
The expression logb((x²y) / (z³)) can be written as (2 × logb(x)) + logb(y) - (3 × logb(z)) using the properties of logarithms of products, quotients, and exponents.
Step-by-step explanation:
The student asked to write logb((x²y) / (z³)) in terms of logarithms of x, y, and z. By using the properties of logarithms, we can simplify this expression step by step.
Firstly, using the property that the logarithm of a division is the difference between the logarithms we write:
logb((x²y) / (z³)) = logb(x²y) - logb(z³)
Next, we apply the property that the logarithm of a product is the sum of the logarithms:
logb(x²y) = logb(x²) + logb(y)
For the terms with exponents, apply the property that the logarithm of a number raised to an exponent is the product of the exponent and the logarithm:
logb(x²) = 2 × logb(x)
logb(z³) = 3 × logb(z)
Combining these, we get the final expression:
logb((x²y) / (z³)) = (2 × logb(x)) + logb(y) - (3 × logb(z))