To expand the expression [tex]\(\log \frac{z}{x^4}\)[/tex] using properties of logarithms, follow these steps:
1. Apply the Logarithm of a Quotient Rule:
The logarithm of a quotient is the logarithm of the numerator minus the logarithm of the denominator. Here, the numerator is [tex]\(z\)[/tex] and the denominator is [tex]\(x^4\)[/tex].
[tex]\[
\log \frac{z}{x^4} = \log(z) - \log(x^4)
\][/tex]
2. Apply the Logarithm of a Power Rule:
The logarithm of a power can be expressed as the exponent times the logarithm of the base. Therefore, [tex]\(\log(x^4)\)[/tex] can be written as:
[tex]\[
\log(x^4) = 4 \log(x)
\][/tex]
3. Combine the Results:
Substitute the expanded form of [tex]\(\log(x^4)\)[/tex] back into the original equation.
[tex]\[
\log(z) - \log(x^4) = \log(z) - 4 \log(x)
\][/tex]
Thus, the expanded form of [tex]\(\log \frac{z}{x^4}\)[/tex] is:
[tex]\[
\log \frac{z}{x^4} = \log(z) - 4 \log(x)
\][/tex]
