Question

What is the fully expanded form of log (x^2/5y)

2 log x - log(5y)
2 log x - log 5 - log y
Log x^2 - log (5y)
Log x - log 5 - log y

Answer 1

To find the fully expanded form of \( \log \left( \frac{x^2}{5y} \right) \), we can use the properties of logarithms. The properties we'll use are:

1. The quotient rule: \( \log \left( \frac{a}{b} \right) = \log a - \log b \)
2. The product rule: \( \log(ab) = \log a + \log b \)
3. The power rule: \( \log(a^n) = n \log a \)

Now we can apply these rules to expand \( \log \left( \frac{x^2}{5y} \right) \):

1. Start with the quotient rule:
\[
\log \left( \frac{x^2}{5y} \right) = \log(x^2) - \log(5y)
\]

2. Now apply the product rule to \( \log(5y) \):
\[
\log(5y) = \log(5) + \log(y)
\]
Therefore:
\[
\log \left( \frac{x^2}{5y} \right) = \log(x^2) - (\log(5) + \log(y))
\]

3. Now apply the power rule to \( \log(x^2) \):
\[
\log(x^2) = 2 \log(x)
\]

4. Putting it all together:
\[
\log \left( \frac{x^2}{5y} \right) = 2 \log(x) - \log(5) - \log(y)
\]

Thus, the fully expanded form is:

\[
2 \log x - \log 5 - \log y
\]

So the correct answer is:
**2 log x - log 5 - log y**.