The `x , y , z` are positive real numbers such that `(log)_(2x)z=3,(log)_(5y)z=6,a n d(log)_(x y)z=2/3,` then the value of `(1/(2z))` is ............

Explanation

To solve the problem, we need to find the value of \( \frac{1}{2z} \) given the logarithmic equations involving \( x, y, z \). Let's break it down step by step.

### Step 1: Convert logarithmic equations to exponential form
We are given three logarithmic equations:
1. \( \log_{2x} z = 3 \)
2. \( \log_{5y} z = 6 \)
3. \( \log_{xy} z = \frac{2}{3} \)

Using the property of logarithms that states \( \log_a b = c \) implies \( b = a^c \), we can rewrite these equations in exponential form:

1. From \( \log_{2x} z = 3 \):
\[ z = (2x)^3 = 8x^3 \]

2. From \( \log_{5y} z = 6 \):
\[ z = (5y)^6 = 15625y^6 \]

3. From \( \log_{xy} z = \frac{2}{3} \):
\[ z = (xy)^{\frac{2}{3}} = x^{\frac{2}{3}} y^{\frac{2}{3}} \]

### Step 2: Set the equations equal to each other
Now we have three expressions for \( z \):
1. \( z = 8x^3 \)
2. \( z = 15625y^6 \)
3. \( z = x^{\frac{2}{3}} y^{\frac{2}{3}} \)

We can set these equations equal to each other to find relationships between \( x \) and \( y \).

### Step 3: Equate the first two expressions for \( z \)
From \( 8x^3 = 15625y^6 \):
\[ \frac{x^3}{y^6} = \frac{15625}{8} \]
This gives us a relationship between \( x \) and \( y \).

### Step 4: Equate the second and third expressions for \( z \)
From \( 15625y^6 = x^{\frac{2}{3}} y^{\frac{2}{3}} \):
\[ \frac{15625y^6}{y^{\frac{2}{3}}} = x^{\frac{2}{3}} \]
This simplifies to:
\[ 15625y^{\frac{16}{3}} = x^{\frac{2}{3}} \]

### Step 5: Substitute \( x \) and \( y \) in terms of \( z \)
From the first equation, we can express \( x \) in terms of \( z \):
\[ x = \left(\frac{z}{8}\right)^{\frac{1}{3}} = \frac{z^{\frac{1}{3}}}{2} \]

From the second equation, we can express \( y \) in terms of \( z \):
\[ y = \left(\frac{z}{15625}\right)^{\frac{1}{6}} = \frac{z^{\frac{1}{6}}}{5} \]

### Step 6: Substitute \( x \) and \( y \) into the third equation
Substituting \( x \) and \( y \) into \( z = x^{\frac{2}{3}} y^{\frac{2}{3}} \):
\[ z = \left(\frac{z^{\frac{1}{3}}}{2}\right)^{\frac{2}{3}} \left(\frac{z^{\frac{1}{6}}}{5}\right)^{\frac{2}{3}} \]

### Step 7: Simplify the equation
This leads to:
\[ z = \frac{z^{\frac{2}{9}}}{4} \cdot \frac{z^{\frac{1}{9}}}{25} \]
Combining the powers of \( z \):
\[ z = \frac{z^{\frac{2}{9} + \frac{1}{9}}}{100} = \frac{z^{\frac{3}{9}}}{100} = \frac{z^{\frac{1}{3}}}{100} \]

### Step 8: Solve for \( z \)
Now we can solve for \( z \):
\[ z^{\frac{2}{3}} = \frac{1}{100} \]
Taking the cube root:
\[ z = \left(\frac{1}{100}\right)^{\frac{3}{2}} = \frac{1}{1000} \]

### Step 9: Find \( \frac{1}{2z} \)
Now we can find \( \frac{1}{2z} \):
\[ \frac{1}{2z} = \frac{1}{2 \cdot \frac{1}{1000}} = \frac{1000}{2} = 500 \]

### Final Answer
Thus, the value of \( \frac{1}{2z} \) is:
\[ \boxed{500} \]