Using the Quotient Rule of Logarithms, the Product Rule of Logarithms, and the Power Rule of Logarithms, how can log(x^2/5y)

Question

be fully expanded?(1 point)
Responses

2logx−log(5y)
2 logarithm x minus logarithm left parenthesis 5 y right parenthesis

logx−log5−logy
logarithm x minus logarithm 5 minus logarithm y

2logx−log5−logy
2 logarithm x minus logarithm 5 minus logarithm y

logx2−log(5y)

Answer 1

To fully expand the expression \(\log\left(\frac{x^2}{5y}\right)\) using the Quotient Rule, Product Rule, and Power Rule of Logarithms, we can follow these steps:

1. Quotient Rule: The logarithm of a quotient is the difference of the logarithms. So, we apply the Quotient Rule first: \[ \log\left(\frac{x^2}{5y}\right) = \log(x^2) - \log(5y) \]

2. Power Rule: The logarithm of a power is the exponent times the logarithm of the base. Use the Power Rule on \(\log(x^2)\): \[ \log(x^2) = 2 \log(x) \] Thus, we rewrite the expression: \[ = 2 \log(x) - \log(5y) \]

3. Product Rule: Now we can expand \(\log(5y)\) using the Product Rule, which states that the logarithm of a product is the sum of the logarithms: \[ \log(5y) = \log(5) + \log(y) \] Substituting this back in gives: \[ = 2 \log(x) - (\log(5) + \log(y)) = 2 \log(x) - \log(5) - \log(y) \]

Putting it all together, we obtain the fully expanded logarithmic expression: \[ 2 \log(x) - \log(5) - \log(y) \]

Among the provided responses, the correct answer is:

2logx - log5 - logy