Question

Using numerical analysis and the Quotient Rule of Logarithms, how can log(r^3/n^2) be rewritten?

Answer 1

To rewrite the expression \( \log\left(\frac{r^3}{n^2}\right) \) using the Quotient Rule of Logarithms, we can apply the following rules of logarithms:

1. The Quotient Rule states that \( \log\left(\frac{a}{b}\right) = \log(a) - \log(b) \).
2. The Power Rule states that \( \log(a^k) = k \cdot \log(a) \).

Now let's apply these rules step by step to rewrite \( \log\left(\frac{r^3}{n^2}\right) \):

1. **Apply the Quotient Rule:**
\[
\log\left(\frac{r^3}{n^2}\right) = \log(r^3) - \log(n^2)
\]

2. **Now apply the Power Rule to both terms:**
- For the first term:
\[
\log(r^3) = 3 \cdot \log(r)
\]
- For the second term:
\[
\log(n^2) = 2 \cdot \log(n)
\]

3. **Combine these results:**
\[
\log\left(\frac{r^3}{n^2}\right) = 3 \cdot \log(r) - 2 \cdot \log(n)
\]

Therefore, the expression \( \log\left(\frac{r^3}{n^2}\right) \) can be rewritten as:

\[
3 \cdot \log(r) - 2 \cdot \log(n)
\]