Which of the following options correctly rewrites ln7+3lnw−2lnx

Question

using the Quotient Rule of Logarithms, the Product Rule of Logarithms, and the Power Rule of Logarithms? (1 point)
Responses

ln(21w2x)
the natural log of open paren 21 w over 2 x close paren

ln(7w3x2)
the natural log of open paren the fraction with numerator 7 w cubed and denominator x squared close paren

ln(73wx)
the natural log of open paren the fraction with numerator 7 and denominator 3 w x close paren

None of these

Answer 1

To rewrite the expression \( \ln 7 + 3 \ln w - 2 \ln x \) using the rules of logarithms, we can apply the Product Rule, Quotient Rule, and Power Rule.

1. Power Rule:
   • \( 3 \ln w = \ln(w^3) \)
   • \( -2 \ln x = \ln(x^{-2}) \)

This lets us rewrite the expression as: \[ \ln 7 + \ln(w^3) + \ln(x^{-2}) \]

2. Product Rule: Now we use the Product Rule, which states that \( \ln a + \ln b = \ln(ab) \). Thus, we have: \[ \ln(7) + \ln(w^3) = \ln(7w^3) \]

3. Now combining this with \( \ln(x^{-2}) \): Using the Product and Quotient Rules together, we have: \[ \ln(7w^3) + \ln(x^{-2}) = \ln\left(\frac{7w^3}{x^2}\right) \]

Now the correct simplified version of the expression is: \[ \ln\left(\frac{7w^3}{x^2}\right) \]

Now let's check the provided options:

1. \( \ln(21w^2x) \)
2. \( \ln(7w^3x^2) \)
3. \( \ln\left(\frac{7}{3wx}\right) \)
4. None of these

The correct expression \( \ln\left(\frac{7w^3}{x^2}\right) \) is not directly present in the options. Hence, the correct response would be: None of these.