If `2^x=4^y=8^z` and `1/(2x)+1/(4y)+1/(4z)=4`, then value of x is :

Options

A. A)`7/(16)`
B. B)`9/(16)`
C. C)`7/(32)`
D. D)`9/(48)`

Answer

To solve the problem step by step, we start with the given equations:

1. **Given Equations**:
\[ 2^x = 4^y = 8^z \]

2. **Expressing in terms of the same base**:
We can express \(4\) and \(8\) in terms of base \(2\):
\[ 4 = 2^2 \quad \text{and} \quad 8 = 2^3 \]
Therefore, we can rewrite the equations as:
\[ 2^x = (2^2)^y = 2^{2y} \quad \text{and} \quad 2^x = (2^3)^z = 2^{3z} \]

3. **Setting the exponents equal**:
Since the bases are the same, we can equate the exponents:
\[ x = 2y \quad \text{and} \quad x = 3z \]

4. **Expressing \(y\) and \(z\) in terms of \(x\)**:
From \(x = 2y\), we can express \(y\) as:
\[ y = \frac{x}{2} \]
From \(x = 3z\), we can express \(z\) as:
\[ z = \frac{x}{3} \]

5. **Substituting into the second equation**:
Now we substitute \(y\) and \(z\) into the equation:
\[ \frac{1}{2x} + \frac{1}{4y} + \frac{1}{4z} = 4 \]
Substituting \(y\) and \(z\):
\[ \frac{1}{2x} + \frac{1}{4 \cdot \frac{x}{2}} + \frac{1}{4 \cdot \frac{x}{3}} = 4 \]

6. **Simplifying the fractions**:
This simplifies to:
\[ \frac{1}{2x} + \frac{1}{2x} + \frac{3}{4x} = 4 \]
Combining the terms:
\[ \frac{1}{2x} + \frac{1}{2x} + \frac{3}{4x} = \frac{2}{2x} + \frac{3}{4x} = \frac{4}{4x} + \frac{3}{4x} = \frac{7}{4x} \]

7. **Setting the equation**:
Now we have:
\[ \frac{7}{4x} = 4 \]

8. **Cross-multiplying**:
Cross-multiplying gives us:
\[ 7 = 16x \]

9. **Solving for \(x\)**:
Finally, solving for \(x\):
\[ x = \frac{7}{16} \]

Thus, the value of \(x\) is:
\[ \boxed{\frac{7}{16}} \]