Which of the following options rewrites ln7+3lnw-2lnx using the Quotient Rule of Logarithms, the Product Rule of Logarithms, and the Power Rule of Logarithms?

Question

1)ln(21w/2x)
2)ln(7w^3/x^2)
3)ln(7/3wx)

Answer 1

To rewrite \( \ln 7 + 3 \ln w - 2 \ln x \) using the laws of logarithms, we can follow these steps:

1. Power Rule: The Power Rule states that \( n \ln a = \ln(a^n) \).
   • For \( 3 \ln w \), we can rewrite this as \( \ln(w^3) \).
   • For \( -2 \ln x \), we can rewrite this as \( \ln(x^{-2}) \).

So we rewrite the expression:

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

2. Product Rule: The Product Rule states that \( \ln a + \ln b = \ln(ab) \).
   • Here we can combine \( \ln 7 \) and \( \ln(w^3) \):

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

3. Combining using the Product Rule again to include \( \ln(x^{-2}) \):

\[ \ln(7w^3) + \ln(x^{-2}) = \ln\left( \frac{7w^3}{x^2} \right) \]

Thus, the expression simplifies to:

\[ \ln\left( \frac{7w^3}{x^2} \right) \]

Therefore, the correct option that rewrites \( \ln 7 + 3 \ln w - 2 \ln x \) is:

2) \( \ln\left( \frac{7w^3}{x^2} \right) \)