Question

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

Answer 1

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

1. **Quotient Rule**: The Quotient Rule states that \( \log\left(\frac{a}{b}\right) = \log(a) - \log(b) \). We can apply this rule to our expression:

\[
\log\left(\frac{x^2}{5^y}\right) = \log(x^2) - \log(5^y)
\]

2. **Power Rule**: The Power Rule states that \( \log(a^b) = b \cdot \log(a) \). We can apply this rule to both \( \log(x^2) \) and \( \log(5^y) \):

- For \( \log(x^2) \):
\[
\log(x^2) = 2 \cdot \log(x)
\]

- For \( \log(5^y) \):
\[
\log(5^y) = y \cdot \log(5)
\]

3. **Substituting Back**: Now we substitute these expanded forms back into the expression:

\[
\log\left(\frac{x^2}{5^y}\right) = 2 \cdot \log(x) - y \cdot \log(5)
\]

So the fully expanded form of \( \log\left(\frac{x^2}{5^y}\right) \) is:

\[
\log\left(\frac{x^2}{5^y}\right) = 2 \cdot \log(x) - y \cdot \log(5)
\]