Question

Which of the following options correctly rewrites ln7+3lnw−2lnx
ln
⁡
7
+
3
ln
⁡
𝑤
−
2
ln
⁡
𝑥
using the Quotient Rule of Logarithms, the Product Rule of Logarithms, and the Power Rule of Logarithms?

Option #1: ​ln(21w2x)
​
ln
⁡
(
21
𝑤
2
𝑥
)

Option #2: ​ln(7w3x2)
​
ln
⁡
(
7
𝑤
3
𝑥
2
)

Option #3: ​ln(73wx)

Answer 1

To simplify the expression \( \ln 7 + 3 \ln w - 2 \ln x \), we will use the properties of logarithms:

1. **Power Rule of Logarithms**: \( n \ln a = \ln(a^n) \)
2. **Product Rule of Logarithms**: \( \ln a + \ln b = \ln(ab) \)
3. **Quotient Rule of Logarithms**: \( \ln a - \ln b = \ln\left(\frac{a}{b}\right) \)

Let's apply these rules step by step:

1. First, we can apply the power rule to \( 3 \ln w \) and \( -2 \ln x \):
\[
3 \ln w = \ln(w^3) \quad \text{and} \quad -2 \ln x = \ln\left(\frac{1}{x^2}\right)
\]

2. Now, we can rewrite the overall expression:
\[
\ln 7 + 3 \ln w - 2 \ln x = \ln 7 + \ln(w^3) + \ln\left(\frac{1}{x^2}\right)
\]

3. Next, we simplify it using the product and quotient rules:
\[
\ln 7 + \ln(w^3) = \ln(7w^3)
\]

Therefore, we have:
\[
\ln(7w^3) + \ln\left(\frac{1}{x^2}\right) = \ln(7w^3) - \ln(x^2) = \ln\left(\frac{7w^3}{x^2}\right)
\]

4. The final result is:
\[
\ln\left(\frac{7w^3}{x^2}\right)
\]

Now let's evaluate the options:

- **Option #1: \(\ln(21w^2x)\)** – This does not match our result.
- **Option #2: \(\ln(7w^3x^2)\)** – This does not match our result.
- **Option #3: \(\ln(73wx)\)** – This does not match our result.

None of the options provided are correct based on our simplified expression. The simplified expression is \(\ln\left(\frac{7w^3}{x^2}\right)\).