Final answer:
The expression log[(x+3)⁴ (x-5)⁷] is expanded using the logarithm rules for products and exponents to become 4 * log(x+3) + 7 * log(x-5).
Step-by-step explanation:
To expand the expression log[(x+3)⁴ (x-5)⁷] using the rules of logarithms, we can apply two main properties. The first property states that the logarithm of a product of two numbers is the sum of the logarithms of the two numbers. The second relevant property is that the logarithm of a number raised to an exponent is the product of the exponent and the logarithm of the number.
Using these properties, we can expand the given logarithmic expression as follows:
- Apply the product rule: log[(x+3)⁴ (x-5)⁷] = log(x+3)⁴ + log(x-5)⁷
- Apply the exponent rule to both terms: = 4 * log(x+3) + 7 * log(x-5)
This results in the fully expanded form of the original expression, which combines the laws of logarithms concerning products and exponents.