Final answer:
Using the power rule and subtraction rule for logarithms, the expression log₄(x²-9) - 6log₄(x+3) can be written as a single logarithm: log₄((x²-9)/(x+3)⁶).
Step-by-step explanation:
To write the expression as a single logarithm log₄(x²-9) - 6log₄(x+3), we can use logarithmic rules, specifically the power rule and the rule for the subtraction of logarithms. First, apply the power rule of logarithms, which states that the logarithm of a number raised to an exponent is equal to the product of the exponent and the logarithm of the number.
In our case, apply the power rule to 6log₄(x+3) to get log₄((x+3)⁶).
Now, using the rule for the subtraction of logarithms, which tells us that the difference of two logarithms is the logarithm of the division of their respective bases, we combine the two logarithms.
This gives us the single logarithm expression log₄((x²-9)/(x+3)⁶).
Remember to check for extraneous solutions when solving for x if this expression were set equal to another value, as logarithms are only defined for positive real numbers.