Final answer:
The difference f(x) - g(x) of the given logarithmic functions simplifies to log2(3), after applying the properties of logarithms.
Step-by-step explanation:
The student has provided two logarithmic functions with the same base, f(x) = log2(3x-9) and g(x) = log2(x-3), and is asking for the difference between these functions, which is f(x) - g(x). Using the properties of logarithms, particularly the one that states the logarithm of the quotient of two numbers is equal to the difference of their logarithms, we can simplify the expression.
Simplified, we have:
f(x) - g(x) = log2(3x-9) - log2(x-3). Applying the property, we get a single logarithm:
log2\((3x-9)/(x-3)\).
Further simplifying the fraction within the logarithm:
log2\(\frac{3(x-3)}{x-3}\) = log2(3), since (x-3) cancels out in the numerator and denominator.
So, the final simplified result for f(x) - g(x) is log2(3), which is just the logarithm of 3 to the base 2.