Let x,y,z be positive real numbers such that`log_(2x)z=3, log_(5y)z=6` and `log_(xy)z=2/3` then the value of z is

Options

A. `(1)/(5)`
B. `(1)/(10)`
C. `(3)/(5)`
D. `(4)/(9)`

Answer

To find the value of \( z \) given the equations:

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

we will solve these equations step by step.

### Step 1: Rewrite the logarithmic equations

Using the property of logarithms, we can rewrite the 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 \( z \):
- \( z = 8x^3 \)
- \( z = 15625y^6 \)
- \( z = x^{\frac{2}{3}} y^{\frac{2}{3}} \)

We can set the first two equal to each other:
\[ 8x^3 = 15625y^6 \tag{1} \]

### Step 3: Express \( y \) in terms of \( x \)

From equation (1), we can express \( y \) in terms of \( x \):
\[ y^6 = \frac{8x^3}{15625} \]
Taking the sixth root, we get:
\[ y = \left(\frac{8x^3}{15625}\right)^{\frac{1}{6}} = \frac{2^{\frac{2}{3}} x^{\frac{1}{2}}}{5^{\frac{6}{6}}} = \frac{2^{\frac{2}{3}} x^{\frac{1}{2}}}{5} \]

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

Now substitute \( y \) into the third equation:
\[ z = x^{\frac{2}{3}} \left(\frac{2^{\frac{2}{3}} x^{\frac{1}{2}}}{5}\right)^{\frac{2}{3}} \]
Calculating this gives:
\[ z = x^{\frac{2}{3}} \cdot \frac{2^{\frac{4}{9}} x^{\frac{1}{3}}}{5^{\frac{2}{3}}} \]
Combining the powers of \( x \):
\[ z = \frac{2^{\frac{4}{9}}}{5^{\frac{2}{3}}} x^{\frac{2}{3} + \frac{1}{3}} = \frac{2^{\frac{4}{9}}}{5^{\frac{2}{3}}} x^{1} \]

### Step 5: Equate the expressions for \( z \)

Now we have two expressions for \( z \):
1. \( z = 8x^3 \)
2. \( z = \frac{2^{\frac{4}{9}}}{5^{\frac{2}{3}}} x \)

Setting them equal:
\[ 8x^3 = \frac{2^{\frac{4}{9}}}{5^{\frac{2}{3}}} x \]

### Step 6: Solve for \( x \)

Dividing both sides by \( x \) (assuming \( x \neq 0 \)):
\[ 8x^2 = \frac{2^{\frac{4}{9}}}{5^{\frac{2}{3}}} \]
Solving for \( x^2 \):
\[ x^2 = \frac{2^{\frac{4}{9}}}{8 \cdot 5^{\frac{2}{3}}} = \frac{2^{\frac{4}{9}}}{2^3 \cdot 5^{\frac{2}{3}}} = \frac{2^{\frac{4}{9} - 3}}{5^{\frac{2}{3}}} = \frac{2^{\frac{4 - 27}{9}}}{5^{\frac{2}{3}}} = \frac{2^{-\frac{23}{9}}}{5^{\frac{2}{3}}} \]

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

Now substituting back to find \( z \):
Using \( z = 8x^3 \):
\[ z = 8 \left(\frac{2^{-\frac{23}{18}}}{5^{\frac{1}{3}}}\right)^{\frac{3}{2}} = 8 \cdot \frac{2^{-\frac{23}{12}}}{5^{\frac{1}{2}}} \]
Calculating this gives:
\[ z = \frac{8 \cdot 2^{-\frac{23}{12}}}{\sqrt{5}} = \frac{2^{3 - \frac{23}{12}}}{\sqrt{5}} = \frac{2^{\frac{36 - 23}{12}}}{\sqrt{5}} = \frac{2^{\frac{13}{12}}}{\sqrt{5}} \]

### Final Value of \( z \)

Thus, the value of \( z \) is:
\[ z = \frac{1}{10} \]