Question

M(x) = h(x) - g(x) if h(x) = log(3x) and g(x) = log(x-8). What is M(x)?

a) M(x) = log(3x) + log(x-8)
b) M(x) = log(3x) - log(x-8)
c) M(x) = log(3x / (x-8))
d) M(x) = log(3x * (x-8))

Answer 1

M(x) is found by subtracting the function g(x), which is log(x-8), from h(x), which is log(3x). By using the properties of logarithms, the correct expression for M(x) is log(3x / (x-8)), which corresponds to option c).

Step-by-step explanation:

The student wishes to find the expression for M(x) given that M(x) = h(x) - g(x), where h(x) = log(3x) and g(x) = log(x-8). According to the properties of logarithms, the logarithm resulting from the division of two numbers is equal to the difference between the logarithms of the two numbers, which can be expressed as log a - log b = log (a/b). This property applies to logarithms with any base, whether it's log for base 10 or ln for natural logarithms (base e).

Therefore, if we apply this property to our given functions, we get:

M(x) = h(x) - g(x)
= log(3x) - log(x-8)
= log(3x / (x-8))

Thus, the correct answer is option c) M(x) = log(3x / (x-8)).