Use the properties of logarithms to expand [tex]\log \frac{z^8}{x}[/tex].

Each logarithm should involve only one variable and should not have any exponents or fractions. Assume that all variables are positive.

[tex]\ \textless \ br/\ \textgreater \ \log \frac{z^8}{x} = \log z^8 - \log x = 8 \log z - \log x\ \textless \ br/\ \textgreater \ [/tex]



Answer :

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]