Sure! To expand [tex]\(\log(x y^3)\)[/tex] using properties of logarithms, you can follow these steps:
1. Product Rule of Logarithms:
The product rule states that the logarithm of a product is the sum of the logarithms of the factors. In mathematical form:
[tex]\[
\log(a \cdot b) = \log(a) + \log(b)
\][/tex]
So, applying the product rule to [tex]\(\log(x y^3)\)[/tex]:
[tex]\[
\log(x y^3) = \log(x) + \log(y^3)
\][/tex]
2. Power Rule of Logarithms:
The power rule states that the logarithm of a number raised to an exponent is the exponent times the logarithm of the base. In mathematical form:
[tex]\[
\log(a^b) = b \cdot \log(a)
\][/tex]
So, applying the power rule to [tex]\(\log(y^3)\)[/tex]:
[tex]\[
\log(y^3) = 3 \cdot \log(y)
\][/tex]
3. Combine Results:
Now, substitute the result from the power rule back into the expanded form from the product rule:
[tex]\[
\log(x y^3) = \log(x) + 3 \cdot \log(y)
\][/tex]
Therefore, the expanded form of [tex]\(\log(x y^3)\)[/tex] is:
[tex]\[
\log(x) + 3 \cdot \log(y)
\][/tex]
