While solving logarithmic equations or logarithmic
inequalities care must be taken to ensure that the value
of the variable obtained do indeed satisfy the given
equation . Often the solution consists in transforming the original equation to form which can be solved with ease .
But in bargain the process the transformations carried
out are not always equivalent .In what follows one must
verify that the values of variables obtained indeed satisfy
original equation or inequation.
How many solutions in real numbers does the
equation ` 3^(x) 8 ^((x)/(x-2)) = 6` have ?
i) None
ii) Exactly one
iii) Exactly two
iv) Infinite

Options

A. None
B. Exactly one
C. Exactly two
D. Infinite

Answer

To solve the equation \( 3^x \cdot 8^{\frac{x}{x-2}} = 6 \), we will follow these steps:

### Step 1: Rewrite the equation
We start with the original equation:
\[ 3^x \cdot 8^{\frac{x}{x-2}} = 6 \]
We can express \( 8 \) as \( 2^3 \):
\[ 3^x \cdot (2^3)^{\frac{x}{x-2}} = 6 \]
This simplifies to:
\[ 3^x \cdot 2^{\frac{3x}{x-2}} = 6 \]

### Step 2: Rewrite the right side
Next, we can express \( 6 \) as \( 3^1 \cdot 2^1 \):
\[ 3^x \cdot 2^{\frac{3x}{x-2}} = 3^1 \cdot 2^1 \]

### Step 3: Compare the bases
Now we can compare the powers of the bases \( 3 \) and \( 2 \):
1. For the base \( 3 \):
\[ x = 1 \]
2. For the base \( 2 \):
\[ \frac{3x}{x-2} = 1 \]

### Step 4: Solve for \( x \) from the second equation
From the equation \( \frac{3x}{x-2} = 1 \), we can cross-multiply:
\[ 3x = x - 2 \]
Rearranging gives:
\[ 3x - x = -2 \implies 2x = -2 \implies x = -1 \]

### Step 5: Verify the solutions
We found two potential solutions: \( x = 1 \) and \( x = -1 \). We need to check if both satisfy the original equation.

1. **Check \( x = 1 \)**:
\[ 3^1 \cdot 8^{\frac{1}{1-2}} = 3 \cdot 8^{-1} = 3 \cdot \frac{1}{8} = \frac{3}{8} \quad \text{(not equal to 6)} \]

2. **Check \( x = -1 \)**:
\[ 3^{-1} \cdot 8^{\frac{-1}{-1-2}} = \frac{1}{3} \cdot 8^{\frac{-1}{-3}} = \frac{1}{3} \cdot 8^{\frac{1}{3}} = \frac{1}{3} \cdot 2 = \frac{2}{3} \quad \text{(not equal to 6)} \]

### Conclusion
Neither \( x = 1 \) nor \( x = -1 \) satisfies the original equation. Therefore, the equation has **no solutions** in real numbers.

### Final Answer
The correct option is:
i) None