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 [tex] \log_b \frac{x}{y} [/tex].

[tex]\[
\begin{aligned}
\log_b \frac{x}{y} & \\
& = \square \checkmark \\
& = \square \checkmark \\
& = \square \checkmark \\
& = \square \checkmark
\end{aligned}
\][/tex]

A. [tex] \log_b x + \log_b y^{-1} [/tex]

B. [tex] \log_b x - \log_b y [/tex]

C. [tex] \log_b x y y^{-1} [/tex]

D. [tex] \log_b x - 1 \log_b y [/tex]



Answer :

Sure, let's break down the process step-by-step by using the power and product properties of logarithms to prove the quotient property for [tex]\(\log_b \frac{x}{y}\)[/tex].

### Step 1:
Start with the given equation:
[tex]\[ \log_b \frac{x}{y} \][/tex]

### Step 2:
Use the power property of logarithms to express the division inside the logarithm as a multiplication with the exponent -1.
[tex]\[ \log_b \frac{x}{y} = \log_b (x \cdot y^{-1}) \][/tex]
Choose the correct option:
C [tex]\(\log_b (x \cdot y^{-1})\)[/tex]

### Step 3:
Express the logarithm of a product as the sum of the logarithms:
[tex]\[ \log_b (x \cdot y^{-1}) = \log_b x + \log_b y^{-1} \][/tex]
Choose the correct option:
A [tex]\(\log_b x + \log_b y^{-1}\)[/tex]

### Step 4:
Use the power property of logarithms again to bring the exponent -1 in front of the logarithm:
[tex]\[ \log_b x + \log_b y^{-1} = \log_b x + (-\log_b y) \][/tex]
Which simplifies to:
[tex]\[ \log_b x - \log_b y \][/tex]
Choose the correct option:
B [tex]\(\log_b x - \log_b y\)[/tex]

### Step 5:
Simplify the expression to its final form:
[tex]\[ \log_b x - \log_b y \][/tex]
Choose the correct option:
B [tex]\(\log_b x - \log_b y\)[/tex]

Putting it all together, we have:
[tex]\[ \begin{aligned} \log_b \frac{x}{y} & = \log_b (x \cdot y^{-1}) \quad \text{(C)} \checkmark \\ & = \log_b x + \log_b y^{-1} \quad \text{(A)} \checkmark \\ & = \log_b x + (-\log_b y) \quad \text{(B)} \checkmark \\ & = \log_b x - \log_b y \quad \text{(B)} \checkmark \end{aligned} \][/tex]

Hence, the final proof that the quotient property is true for [tex]\(\log_b \frac{x}{y}\)[/tex] is complete.