To expand the expression [tex]\(\log \frac{z^8}{x}\)[/tex] using the properties of logarithms, follow these steps:
1. Identify the Components: Recognize that the given expression involves a logarithm of a quotient:
[tex]\[
\log \frac{z^8}{x}
\][/tex]
2. Apply the Quotient Rule: The logarithm of a quotient [tex]\(\log \left( \frac{a}{b} \right)\)[/tex] can be expanded as the difference of two logarithms:
[tex]\[
\log \left( \frac{a}{b} \right) = \log a - \log b
\][/tex]
Applying this rule to our expression, we get:
[tex]\[
\log \frac{z^8}{x} = \log z^8 - \log x
\][/tex]
3. Apply the Power Rule: The logarithm of a power [tex]\(\log (a^b)\)[/tex] can be expanded as the exponent times the logarithm of the base:
[tex]\[
\log (a^b) = b \log a
\][/tex]
Applying this rule to [tex]\(\log z^8\)[/tex], we get:
[tex]\[
\log z^8 = 8 \log z
\][/tex]
Now, substitute this back into the expanded expression from step 2:
[tex]\[
\log \frac{z^8}{x} = 8 \log z - \log x
\][/tex]
Thus, the expanded form of [tex]\(\log \frac{z^8}{x}\)[/tex] is:
[tex]\[
\log \frac{z^8}{x} = 8 \log z - \log x
\][/tex]
So, the completed expression is:
[tex]\[
\log \frac{z^8}{x} = 8 \log z - \log x
\][/tex]
