Using numerical analysis and the Quotient Rule of Logarithms, how can log(r3/n2)

Question

be rewritten?(1 point)
Responses

2logn−3logr
2 logarithm n minus 3 logarithm r

3logr/2logn
Start Fraction 3 logarithm r over 2 logarithm n End Fraction

3logr−2logn
3 logarithm r minus 2 logarithm n

3logr+2logn

Answer 1

To rewrite \( \log\left(\frac{r^3}{n^2}\right) \) using the Quotient Rule of Logarithms and other logarithmic properties, we can follow these steps:

1. Apply the Quotient Rule of Logarithms: \[ \log\left(\frac{a}{b}\right) = \log(a) - \log(b) \] Thus, \[ \log\left(\frac{r^3}{n^2}\right) = \log(r^3) - \log(n^2) \]

2. Next, apply the Power Rule of Logarithms: \[ \log(a^b) = b \cdot \log(a) \] Therefore, \[ \log(r^3) = 3 \cdot \log(r) \quad \text{and} \quad \log(n^2) = 2 \cdot \log(n) \]

3. Substitute these results back into the expression: \[ \log\left(\frac{r^3}{n^2}\right) = 3 \cdot \log(r) - 2 \cdot \log(n) \]

Putting it all together, we get: \[ \log\left(\frac{r^3}{n^2}\right) = 3\log(r) - 2\log(n) \]

Therefore, the correct response is: 3logr−2logn (3 logarithm r minus 2 logarithm n).