Question

Assuming x and y are positive, use properties of logarithms to write the expression as a sum or difference of logarithms or multiples of logarithms. log xy^3

Answer 1

To express \( \log xy^3 \) as a sum or difference of logarithms, we can use the properties of logarithms. Here are the relevant properties:

1. **Product Property**: \( \log(ab) = \log a + \log b \)
2. **Power Property**: \( \log(a^n) = n \log a \)

Using these properties, we can transform \( \log xy^3 \):

1. Apply the Product Property to separate the logarithm of the product:
\[
\log xy^3 = \log x + \log y^3
\]

2. Next, apply the Power Property to simplify \( \log y^3 \):
\[
\log y^3 = 3 \log y
\]

Putting it all together, we have:
\[
\log xy^3 = \log x + \log y^3 = \log x + 3 \log y
\]

So the final expression as a sum of logarithms is:
\[
\log xy^3 = \log x + 3 \log y
\]