If `u = v^(2) = w^(2) =z^(4)`, then `log_(u)(uvwz)`, is equal to

Options

A. `1+1/2 -1/3 - 1/4`
B. 24
C. `1+1/2 + 1/3 + 1/4`
D. `1/24`

Answer

To solve the problem where \( u = v^2 = w^2 = z^4 \) and we need to find \( \log_u(uvwz) \), we can follow these steps:

### Step 1: Express \( v, w, z \) in terms of \( u \)
Given:
- \( u = v^2 \) implies \( v = u^{1/2} \)
- \( u = w^2 \) implies \( w = u^{1/2} \)
- \( u = z^4 \) implies \( z = u^{1/4} \)

### Step 2: Substitute \( v, w, z \) into \( uvwz \)
Now, substituting \( v, w, z \) into \( uvwz \):
\[ uvwz = u \cdot v \cdot w \cdot z = u \cdot (u^{1/2}) \cdot (u^{1/2}) \cdot (u^{1/4}) \]

### Step 3: Combine the powers of \( u \)
Combining the powers of \( u \):
\[ uvwz = u^{1 + 1/2 + 1/2 + 1/4} \]
Calculating the exponent:
\[ 1 + 1/2 + 1/2 + 1/4 = 1 + 1 = 2 + 1/4 = 2.25 = \frac{9}{4} \]
Thus,
\[ uvwz = u^{\frac{9}{4}} \]

### Step 4: Calculate \( \log_u(uvwz) \)
Now, we can calculate \( \log_u(uvwz) \):
\[ \log_u(uvwz) = \log_u(u^{\frac{9}{4}}) \]
Using the property of logarithms \( \log_b(b^x) = x \):
\[ \log_u(u^{\frac{9}{4}}) = \frac{9}{4} \]

### Final Answer
Thus, the value of \( \log_u(uvwz) \) is:
\[ \frac{9}{4} \]

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