Final answer:
To expand the expression log₆(36xy²), use the properties of logarithms to break it down into 2 (from log₆(6²)), log₆(x), and 2·log₆(y) from log₆(y²), resulting in 2 + log₆(x) + 2·log₆(y) as the final expanded form.
Step-by-step explanation:
The question is asking to expand the logarithmic expression log₆(36xy²). To expand this logarithmic expression, we apply the properties of logarithms, such as the product rule, which states that the logarithm of a product is equal to the sum of the logarithms of the individual factors.
To expand log₆(36xy²), let's break it down:
- First, recognize that 36 is a perfect square, specifically, it's 6².
- According to the power rule of logarithms, we can take the exponent and multiply it with the logarithm. Therefore, log₆(6²) simplifies to 2·log₆(6).
- Since log₆(6) equals 1 (because the base and the argument are the same), we can simplify further to get 2.
- The expression log₆(xy²) can be expanded as log₆(x) + log₆(y²) using the product rule.
- Applying the power rule to log₆(y²), it becomes 2·log₆(y).
Putting it all together, the expanded form is:
2 + log₆(x) + 2·log₆(y)