A line `x=lamda` intersects the graph of `y=log_5`x and `y=log_5(x+4)`.The distance between the point of intersection is 0.5. Given `lamda=a+sqrtb`,where a and b are integers , the value of (a+b) is

Answer

To solve the problem, we need to find the value of \( \lambda \) where the line \( x = \lambda \) intersects the graphs of \( y = \log_5 x \) and \( y = \log_5(x + 4) \). The distance between the points of intersection is given as 0.5.

### Step-by-step Solution:

1. **Set Up the Equations**:
The points of intersection can be represented as:
\[ y_1 = \log_5(\lambda) \quad \text{and} \quad y_2 = \log_5(\lambda + 4) \]

2. **Distance Between Points**:
The distance between the two points of intersection is given by:
\[ |y_2 - y_1| = 0.5 \]
This can be expressed as:
\[ \left| \log_5(\lambda + 4) - \log_5(\lambda) \right| = 0.5 \]

3. **Using Logarithm Properties**:
By the properties of logarithms, we can rewrite the expression:
\[ \log_5\left(\frac{\lambda + 4}{\lambda}\right) = 0.5 \]
This means:
\[ \frac{\lambda + 4}{\lambda} = 5^{0.5} = \sqrt{5} \]

4. **Cross-Multiplying**:
Rearranging the equation gives:
\[ \lambda + 4 = \sqrt{5} \lambda \]
Rearranging further, we have:
\[ 4 = \sqrt{5} \lambda - \lambda \]
\[ 4 = (\sqrt{5} - 1) \lambda \]

5. **Solving for \( \lambda \)**:
Thus, we can solve for \( \lambda \):
\[ \lambda = \frac{4}{\sqrt{5} - 1} \]

6. **Rationalizing the Denominator**:
To simplify \( \lambda \), we rationalize the denominator:
\[ \lambda = \frac{4(\sqrt{5} + 1)}{(\sqrt{5} - 1)(\sqrt{5} + 1)} = \frac{4(\sqrt{5} + 1)}{5 - 1} = \frac{4(\sqrt{5} + 1)}{4} = \sqrt{5} + 1 \]

7. **Identifying \( a \) and \( b \)**:
From the expression \( \lambda = 1 + \sqrt{5} \), we can identify:
\[ a = 1 \quad \text{and} \quad b = 5 \]

8. **Calculating \( a + b \)**:
Finally, we find:
\[ a + b = 1 + 5 = 6 \]

### Final Answer:
The value of \( a + b \) is \( 6 \).