The area bounded by the curves `y=logx,y=2^(x)` and the lines `x=(1)/(2),x=2` is

Answer

To find the area bounded by the curves \( y = \log x \), \( y = 2^x \), and the lines \( x = \frac{1}{2} \) and \( x = 2 \), we follow these steps:

### Step 1: Find the Intersection Points
We need to find the points where the curves intersect. This requires solving the equation:
\[ \log x = 2^x \]

1. **Evaluate at \( x = \frac{1}{2} \)**:
\[ y = \log\left(\frac{1}{2}\right) = -\log 2 \]
\[ y = 2^{\frac{1}{2}} = \sqrt{2} \]

2. **Evaluate at \( x = 2 \)**:
\[ y = \log(2) \]
\[ y = 2^2 = 4 \]

### Step 2: Set Up the Integral for Area
The area \( A \) between the curves from \( x = \frac{1}{2} \) to \( x = 2 \) can be calculated using the integral:
\[ A = \int_{\frac{1}{2}}^{2} (2^x - \log x) \, dx \]

### Step 3: Calculate the Integral
We will compute the integral:
\[ A = \int_{\frac{1}{2}}^{2} 2^x \, dx - \int_{\frac{1}{2}}^{2} \log x \, dx \]

1. **Calculate \( \int 2^x \, dx \)**:
\[ \int 2^x \, dx = \frac{2^x}{\log 2} \]
Evaluate from \( \frac{1}{2} \) to \( 2 \):
\[ \left[ \frac{2^x}{\log 2} \right]_{\frac{1}{2}}^{2} = \frac{2^2}{\log 2} - \frac{2^{\frac{1}{2}}}{\log 2} = \frac{4 - \sqrt{2}}{\log 2} \]

2. **Calculate \( \int \log x \, dx \)** using integration by parts:
Let \( u = \log x \) and \( dv = dx \), then \( du = \frac{1}{x} dx \) and \( v = x \).
\[ \int \log x \, dx = x \log x - x \]
Evaluate from \( \frac{1}{2} \) to \( 2 \):
\[ \left[ x \log x - x \right]_{\frac{1}{2}}^{2} = \left( 2 \log 2 - 2 \right) - \left( \frac{1}{2} \log \frac{1}{2} - \frac{1}{2} \right) \]
Simplifying:
\[ = 2 \log 2 - 2 - \left( -\frac{1}{2} \log 2 + \frac{1}{2} \right) = 2 \log 2 - 2 + \frac{1}{2} \log 2 - \frac{1}{2} \]
\[ = \frac{5}{2} \log 2 - \frac{5}{2} \]

### Step 4: Combine Results
Now substitute back into the area formula:
\[ A = \left( \frac{4 - \sqrt{2}}{\log 2} \right) - \left( \frac{5}{2} \log 2 - \frac{5}{2} \right) \]
Simplifying:
\[ A = \frac{4 - \sqrt{2}}{\log 2} + \frac{5}{2} - \frac{5}{2} \log 2 \]

### Step 5: Final Area Expression
The final area bounded by the curves is:
\[ A = \frac{4 - \sqrt{2}}{\log 2} + \frac{5}{2} - \frac{5}{2} \log 2 \]