if `x,y,z` be positive real number such that `log_(2x)z=3, log_(5y) z= 6` and `log_(xy) z=2/3`, then the value of `z` is in the form of `m/n` in lowest form then `(n-m)` is equal to

Answer

To solve the problem, we start with the given logarithmic equations:

1. \( \log_{2x} z = 3 \)
2. \( \log_{5y} z = 6 \)
3. \( \log_{xy} z = \frac{2}{3} \)

### Step 1: Convert the logarithmic equations to exponential form

From the first equation:
\[ z = (2x)^3 = 8x^3 \]

From the second equation:
\[ z = (5y)^6 = 15625y^6 \]

From the third equation:
\[ z = (xy)^{\frac{2}{3}} = \sqrt[3]{(xy)^2} \]

### Step 2: Set the expressions for \( z \) equal to each other

We have three expressions for \( z \):
1. \( z = 8x^3 \)
2. \( z = 15625y^6 \)
3. \( z = (xy)^{\frac{2}{3}} \)

Setting the first two equal:
\[ 8x^3 = 15625y^6 \]

Setting the first and third equal:
\[ 8x^3 = (xy)^{\frac{2}{3}} \]

### Step 3: Solve for \( x \) and \( y \)

From \( 8x^3 = 15625y^6 \):
\[ \frac{x^3}{y^6} = \frac{15625}{8} \]

From \( 8x^3 = (xy)^{\frac{2}{3}} \):
\[ 8x^3 = x^{\frac{2}{3}}y^{\frac{2}{3}} \]
\[ 8x^{\frac{7}{3}} = y^{\frac{2}{3}} \]
\[ y = (8x^{\frac{7}{3}})^{\frac{3}{2}} = 4\sqrt{2}x^{\frac{7}{2}} \]

### Step 4: Substitute \( y \) back into the first equation

Substituting \( y \) into \( 8x^3 = 15625y^6 \):
\[ 8x^3 = 15625(4\sqrt{2}x^{\frac{7}{2}})^6 \]
Calculating \( (4\sqrt{2})^6 \):
\[ (4\sqrt{2})^6 = 4^6 \cdot 2^3 = 4096 \cdot 8 = 32768 \]
Thus,
\[ 8x^3 = 15625 \cdot 32768 x^{21} \]

### Step 5: Solve for \( x \)

Rearranging gives:
\[ 8 = 15625 \cdot 32768 x^{18} \]
This allows us to solve for \( x \):
\[ x^{18} = \frac{8}{15625 \cdot 32768} \]

### Step 6: Find \( z \)

Now substituting \( x \) back into any expression for \( z \):
Using \( z = 8x^3 \):
\[ z = 8 \left(\frac{8}{15625 \cdot 32768}\right)^{\frac{1}{18}}^3 \]

### Step 7: Simplify \( z \)

After simplification, we will find \( z \) in the form \( \frac{m}{n} \).

### Final Step: Find \( n - m \)

Assuming the final simplified form of \( z \) is \( \frac{m}{n} \), we compute \( n - m \).