Final answer:
To condense the expression log₄ (x+8) - 4log₄x, we employ the power rule to rewrite 4log₄x as log₄(x^4) and then use the subtraction rule to express the difference as log₄((x+8)/x^4).
Step-by-step explanation:
To condense the expression log₄ (x+8) - 4log₄x using the properties of logarithms, we need to apply various logarithm rules such as the logarithm of a quotient, the logarithm of a power, and the logarithm of a product.
Step 1: Apply the power rule of logarithms
The power rule states that the logarithm of a number raised to an exponent is the product of the exponent and the logarithm of the number. We can rewrite 4log₄x as log₄(x4).
Step 2: Apply the subtraction rule of logarithms
The subtraction rule tells us that the logarithm of the number resulting from the division of two numbers is the difference between the logarithms of the two numbers. Therefore, log₄ (x+8) - log₄(x4) can be expressed as log₄ ((x+8)/x4).
Following these steps, we can condense the given logarithmic expression to log₄ ((x+8)/x4).