Final answer:
The correct expression of 1/3 * log(x) - log(5) as a single logarithm is log(x/5) after applying the power rule and quotient rule of logarithms, making the right choice b) log(x/5).
Step-by-step explanation:
The student is asking to express 1/3 * log(x) - log(5) as a single logarithm. To solve this, we will use the properties of logarithms which state that the logarithm of a number raised to an exponent is the product of the exponent and the logarithm of the number, and the logarithm of the division of two numbers is the difference between the logarithms of the two numbers. First, we will use the power rule of logarithms which can be stated as log(a^b) = b * log(a), to rewrite 1/3 * log(x) as log(x1/3).
Next, we apply the quotient rule, which is log(a) - log(b) = log(a/b), to combine log(x1/3) - log(5) into a single logarithm log(x1/3/5). Since x1/3 is the cube root of x, this can be expressed as x/5 raised to the 1/3 power, which yields log((x/5)1/3). However, the cube root of 5 is not the same as the number 15, as can incorrectly be assumed from option a).
The correct expression of 1/3 * log(x) - log(5) is log((x/5)1/3), which simplifies to log(x/5) since raising a number to the 1/3 power and multiplying it by itself three times would give the original number. Thus, the correct answer is b) log(x/5).