Question

choose the letter of the expression listed on the right that completes each step to show how to use the power and product properties of logarithms to prove that the quotient property is true for \\(\log_b \frac{x}{y}\\).

\\(\log_b \frac{x}{y}\\)
\\(= \quad\\)
\\(= \quad\\)
\\(= \quad\\)
\\(= \quad\\)

a \\(\log_b x + \log_b y^{-1}\\)
b \\(\log_b x - \log_b y\\)
c \\(\log_b x y^{-1}\\)
d \\(\log_b x - 1 \log_b y\\)

Explanation:

⚡ Using what you learned: properties of logarithms

Step 1: Rewrite the quotient using a negative exponent

$$ \log_b \frac{x}{y} = \log_b (x \cdot y^{-1}) = \log_b xy^{-1} $$

This matches option C.

Step 2: Apply the product property of logarithms

$$ \log_b xy^{-1} = \log_b x + \log_b y^{-1} $$

This matches option A.

Step 3: Apply the power property of logarithms to the second term

$$ \log_b x + \log_b y^{-1} = \log_b x + (-1)\log_b y = \log_b x - 1\log_b y $$

This matches option D.

Step 4: Simplify the expression

$$ \log_b x - 1\log_b y = \log_b x - \log_b y $$

This matches option B.

Answer:

The correct order of the expressions from top to bottom is:

1. C (\(\log_b xy^{-1}\))
2. A (\(\log_b x + \log_b y^{-1}\))
3. D (\(\log_b x - 1\log_b y\))
4. B (\(\log_b x - \log_b y\))