Final answer:
To combine the logarithmic expression into a single logarithm, we use the power rule to account for the exponents, the product rule to combine the terms with addition, and the quotient rule to take the difference of the terms with subtraction. This simplifies down to A = x*(2y)^4 / (2z)^4.
Step-by-step explanation:
To rewrite the expression log 2x + 4log 2y - 4log 2z as a single logarithm log(2A), we can use the properties of logarithms, specifically the product, quotient, and power rules of exponents within logarithms. We start with:
log 2x + 4log 2y - 4log 2z
- Apply the power rule (the logarithm of a number raised to an exponent is the product of the exponent and the logarithm of the number):
log 2x + log (2y)^4 - log (2z)^4 - Use the product rule (the logarithm of a product of two numbers is the sum of the logarithms):
log (2x*(2y)^4) - Use the quotient rule (the logarithm resulting from the division of two numbers is the difference between their logarithms):
log (2x*(2y)^4 / (2z)^4)
Considering the properties highlighted, the function A is obtained by simplifying the inside of the final logarithm:
A = x*(2y)^4 / (2z)^4