Sure! Let's start from the logarithmic expression [tex]\(\log \left(y^4 x\right)\)[/tex] and use the properties of logarithms to expand it.
First, recall that the logarithm of a product can be expressed as the sum of the logarithms of the factors:
[tex]\[
\log(A \cdot B) = \log(A) + \log(B)
\][/tex]
In our case, we can treat [tex]\(y^4\)[/tex] and [tex]\(x\)[/tex] as the factors:
[tex]\[
\log \left(y^4 x\right) = \log(y^4) + \log(x)
\][/tex]
Next, use the property of logarithms that states that the logarithm of a power can be expressed as the exponent times the logarithm of the base:
[tex]\[
\log(A^B) = B \cdot \log(A)
\][/tex]
Applying this property to [tex]\(\log(y^4)\)[/tex], we get:
[tex]\[
\log(y^4) = 4 \cdot \log(y)
\][/tex]
Combining these results together, we can write the expanded form of the original expression:
[tex]\[
\log \left(y^4 x\right) = 4 \cdot \log(y) + \log(x)
\][/tex]
Thus, the expansion of [tex]\(\log \left(y^4 x\right)\)[/tex] is:
[tex]\[
4 \cdot \log(y) + \log(x)
\][/tex]
