To expand the expression [tex]\(\log \frac{x^4}{y}\)[/tex] using the properties of logarithms, we can follow these steps:
1. Property of Logarithms - Quotient Rule: The logarithm of a quotient is the difference of the logarithms. In this case:
[tex]\[\log \frac{x^4}{y} = \log(x^4) - \log(y)\][/tex]
2. Property of Logarithms - Power Rule: The logarithm of a power is the exponent times the logarithm of the base. So, we apply this property to [tex]\(\log(x^4)\)[/tex]:
[tex]\[\log(x^4) = 4 \log(x)\][/tex]
3. Combining the results from the steps above, we have:
[tex]\[\log \frac{x^4}{y} = 4 \log(x) - \log(y)\][/tex]
So the fully expanded form is:
[tex]\[
\log \frac{x^4}{y} = 4 \log(x) - 1 \log(y)
\][/tex]
