The solution of the equation `int_(log_(2))^(x) (1)/(e^(x)-1)dx=log(3)/(2)` is given by x=

Options

A. `e^(2)`
B. `1//e`
C. log 4
D. none of these

Answer

To solve the equation

\[ \int_{\log 2}^{x} \frac{1}{e^{x}-1} \, dx = \frac{\log 3}{2}, \]

we will follow these steps:

### Step 1: Set up the integral
We start with the integral given in the equation:

\[ I = \int_{\log 2}^{x} \frac{1}{e^{t}-1} \, dt. \]

### Step 2: Change of variables
Let us make the substitution \( e^{t} - 1 = u \). Then, we have:

\[ e^{t} = u + 1 \quad \text{and} \quad dt = \frac{du}{e^{t}} = \frac{du}{u + 1}. \]

### Step 3: Change the limits of integration
When \( t = \log 2 \):

\[ u = e^{\log 2} - 1 = 2 - 1 = 1. \]

When \( t = x \):

\[ u = e^{x} - 1. \]

Thus, the integral becomes:

\[ I = \int_{1}^{e^{x}-1} \frac{1}{u} \cdot \frac{1}{u + 1} \, du. \]

### Step 4: Simplify the integral using partial fractions
We can express \( \frac{1}{u(u + 1)} \) using partial fractions:

\[ \frac{1}{u(u + 1)} = \frac{1}{u} - \frac{1}{u + 1}. \]

### Step 5: Integrate
Now we can integrate:

\[ I = \int_{1}^{e^{x}-1} \left( \frac{1}{u} - \frac{1}{u + 1} \right) \, du. \]

This gives us:

\[ I = \left[ \log u - \log(u + 1) \right]_{1}^{e^{x}-1}. \]

### Step 6: Evaluate the limits
Evaluating the integral at the limits:

\[ I = \left( \log(e^{x}-1) - \log(e^{x}) \right) - \left( \log(1) - \log(2) \right). \]

Since \( \log(1) = 0 \), we have:

\[ I = \log(e^{x}-1) - x + \log(2). \]

### Step 7: Set the equation to \(\frac{\log 3}{2}\)
Now we set this equal to \(\frac{\log 3}{2}\):

\[ \log(e^{x}-1) - x + \log(2) = \frac{\log 3}{2}. \]

### Step 8: Rearranging the equation
Rearranging gives:

\[ \log(e^{x}-1) = x - \log(2) + \frac{\log 3}{2}. \]

### Step 9: Exponentiate both sides
Exponentiating both sides results in:

\[ e^{x}-1 = e^{x - \log(2) + \frac{\log 3}{2}}. \]

### Step 10: Solve for \(x\)
This simplifies to:

\[ e^{x} - 1 = \frac{3}{2} \cdot \frac{e^{x}}{2}. \]

Multiplying through by 2 gives:

\[ 2e^{x} - 2 = 3e^{x} \implies e^{x} = 2. \]

Taking the natural logarithm:

\[ x = \log 2. \]

### Final Answer
Thus, the solution for \(x\) is:

\[ \boxed{\log 4}. \]