If `log_2x+log_x2=10/3=log_2y+log_y2` and `x!=y` ,then `x+y=`

Options

A. 2
B. `65//8`
C. `37//6`
D. none of these

Explanation

To solve the equation \( \log_2 x + \log_x 2 = \frac{10}{3} = \log_2 y + \log_y 2 \) with the condition \( x \neq y \), we can follow these steps:

### Step 1: Rewrite the logarithmic equations
We start with the equation:
\[ \log_2 x + \log_x 2 = \frac{10}{3} \]
Using the change of base formula, we can rewrite \( \log_x 2 \) as \( \frac{1}{\log_2 x} \):
\[ \log_2 x + \frac{1}{\log_2 x} = \frac{10}{3} \]

### Step 2: Let \( t = \log_2 x \)
Let \( t = \log_2 x \). Then we can rewrite the equation as:
\[ t + \frac{1}{t} = \frac{10}{3} \]

### Step 3: Multiply through by \( t \)
To eliminate the fraction, multiply both sides by \( t \):
\[ t^2 + 1 = \frac{10}{3} t \]

### Step 4: Rearrange the equation
Rearranging gives us:
\[ 3t^2 - 10t + 3 = 0 \]

### Step 5: Solve the quadratic equation
Now we can use the quadratic formula \( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 3, b = -10, c = 3 \):
\[ t = \frac{10 \pm \sqrt{(-10)^2 - 4 \cdot 3 \cdot 3}}{2 \cdot 3} \]
\[ t = \frac{10 \pm \sqrt{100 - 36}}{6} \]
\[ t = \frac{10 \pm \sqrt{64}}{6} \]
\[ t = \frac{10 \pm 8}{6} \]

### Step 6: Calculate the values of \( t \)
This gives us two possible values for \( t \):
1. \( t = \frac{18}{6} = 3 \)
2. \( t = \frac{2}{6} = \frac{1}{3} \)

### Step 7: Find \( x \) from \( t \)
Now we can find \( x \):
- For \( t = 3 \):
\[ \log_2 x = 3 \implies x = 2^3 = 8 \]
- For \( t = \frac{1}{3} \):
\[ \log_2 x = \frac{1}{3} \implies x = 2^{1/3} \]

### Step 8: Repeat for \( y \)
We apply the same process for \( y \) using the equation \( \log_2 y + \log_y 2 = \frac{10}{3} \). Following the same steps, we will find that:
- For \( y \) corresponding to \( t = 3 \), \( y = 8 \)
- For \( y \) corresponding to \( t = \frac{1}{3} \), \( y = 2^{1/3} \)

### Step 9: Determine \( x + y \)
Since \( x \) and \( y \) must be different, we take:
\[ x = 8 \quad \text{and} \quad y = 2^{1/3} \]
Thus,
\[ x + y = 8 + 2^{1/3} \]

### Final Answer
\[ x + y = 8 + 2^{1/3} \]

---