IDNLearn.com provides a user-friendly platform for finding answers to your questions. Discover trustworthy solutions to your questions quickly and accurately with help from our dedicated community of experts.

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]


Sagot :

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]