Certainly! Let's use the Laws of Logarithms to expand the expression [tex]\(\log \left(\sqrt[4]{x^2+y^5}\right)\)[/tex] in a detailed, step-by-step manner.
### Step-by-Step Solution
#### Step 1: Understand the given expression
The given expression is:
[tex]\[ \log \left(\sqrt[4]{x^2 + y^5}\right) \][/tex]
#### Step 2: Recognize the form of the inner expression
The inner expression is [tex]\(\sqrt[4]{x^2 + y^5}\)[/tex], which can be rewritten using exponents:
[tex]\[ \sqrt[4]{x^2 + y^5} = (x^2 + y^5)^{1/4} \][/tex]
#### Step 3: Apply the power rule of logarithms
The power rule of logarithms states that [tex]\(\log(a^b) = b \cdot \log(a)\)[/tex]. Here, our [tex]\(a\)[/tex] is [tex]\(x^2 + y^5\)[/tex] and [tex]\(b\)[/tex] is [tex]\(1/4\)[/tex]. Applying this rule, we get:
[tex]\[ \log \left((x^2 + y^5)^{1/4}\right) = \frac{1}{4} \cdot \log(x^2 + y^5) \][/tex]
#### Step 4: Write the expanded form
Putting it all together, our expanded expression is:
[tex]\[ \log \left(\sqrt[4]{x^2 + y^5}\right) = \frac{1}{4} \log (x^2 + y^5) \][/tex]
### Final Answer
Using the Laws of Logarithms, we have expanded the given logarithmic expression as follows:
[tex]\[ \log \left(\sqrt[4]{x^2 + y^5}\right) = \frac{1}{4} \log (x^2 + y^5) \][/tex]
