Question

Which of the following options correctly rewrites ln7+3lnw−2lnx using the Quotient Rule of Logarithms, the Product Rule of Logarithms, and the Power Rule of Logarithms? Option #1: ​ln(21w2x) Option #2: ​ln(7w3x2) Option #3: ​ln(73wx)

Answer 1

To combine the expression \( \ln 7 + 3 \ln w - 2 \ln x \) using the rules of logarithms, we will apply the following:

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

Let's work through the expression:

1. Start with the given expression:
\[
\ln 7 + 3 \ln w - 2 \ln x
\]

2. Apply the Power Rule to \( 3 \ln w \) and \( -2 \ln x \):
- \( 3 \ln w = \ln(w^3) \)
- \( -2 \ln x = \ln(x^{-2}) \)

The expression becomes:
\[
\ln 7 + \ln(w^3) + \ln(x^{-2})
\]

3. Now, combine the logarithms using the Product Rule:
\[
\ln(7) + \ln(w^3) = \ln(7w^3)
\]
So now we have:
\[
\ln(7w^3) + \ln(x^{-2})
\]

4. Use the Quotient Rule:
\[
\ln(7w^3) + \ln(x^{-2}) = \ln\left(\frac{7w^3}{x^2}\right)
\]

Thus, the final expression is:
\[
\ln\left(\frac{7w^3}{x^2}\right)
\]

None of the options appear to directly present this final expression. However, let's look at the individual components contained in each of the options:

- **Option #1**: \( \ln(21w^2x) \)
- **Option #2**: \( \ln(7w^3x^2) \)
- **Option #3**: \( \ln(73wx) \)

Comparing \( \frac{7w^3}{x^2} \):
None of the expressions match \( \ln\left(\frac{7w^3}{x^2}\right) \), which indicates none of the given options are correct.

The correct simplification is \( \ln\left(\frac{7w^3}{x^2}\right) \).