Sure, let's expand the logarithmic expression [tex]\(\log \frac{y^5}{x}\)[/tex] step-by-step using the properties of logarithms.
### Step 1: Apply the Quotient Rule
The logarithm of a quotient can be expressed as the difference of the logarithms. Specifically, [tex]\(\log \frac{a}{b} = \log a - \log b\)[/tex].
Applying this property, we get:
[tex]\[\log \frac{y^5}{x} = \log(y^5) - \log(x).\][/tex]
### Step 2: Apply the Power Rule
The logarithm of a power can be expressed as the exponent times the logarithm of the base. Specifically, [tex]\(\log(a^b) = b \log(a)\)[/tex].
Applying this property to the term [tex]\(\log(y^5)\)[/tex], we get:
[tex]\[\log(y^5) = 5 \log(y).\][/tex]
### Step 3: Substitute Back
Substitute the expanded form of [tex]\(\log(y^5)\)[/tex] back into the expression from Step 1:
[tex]\[\log \frac{y^5}{x} = 5 \log(y) - \log(x).\][/tex]
### Conclusion
The expanded form of [tex]\(\log \frac{y^5}{x}\)[/tex] is:
[tex]\[ \log \frac{y^5}{x} = 5 \log(y) - \log(x). \][/tex]
So, our final expression is:
[tex]\[ \boxed{5 \log(y) - \log(x).} \][/tex]
This neatly expands the original logarithmic expression, involving each variable in a separate logarithm without any exponents inside the logarithm.
