If `x = 1 + log 2 - log 5, y = 2 log 3 and z = log a - log 5`, find the value of a, if x + y = 2z.

Answer

To solve the problem, we need to find the value of \( a \) given the equations for \( x \), \( y \), and \( z \) and the condition \( x + y = 2z \).

### Step-by-Step Solution:

1. **Write down the given equations:**
\[ x = 1 + \log 2 - \log 5 \]
\[ y = 2 \log 3 \]
\[ z = \log a - \log 5 \]

2. **Set up the equation based on the condition \( x + y = 2z \):**
\[ (1 + \log 2 - \log 5) + (2 \log 3) = 2(\log a - \log 5) \]

3. **Simplify the left-hand side:**
\[ 1 + \log 2 - \log 5 + 2 \log 3 = 1 + \log 2 + \log 9 - \log 5 \]
(since \( 2 \log 3 = \log 3^2 = \log 9 \))

4. **Rewrite \( 1 \) as \( \log 10 \):**
\[ \log 10 + \log 2 + \log 9 - \log 5 \]

5. **Combine the logarithms on the left-hand side:**
Using the property \( \log m + \log n = \log(m \cdot n) \):
\[ \log \left(10 \cdot 2 \cdot 9\right) - \log 5 = \log \left(\frac{10 \cdot 2 \cdot 9}{5}\right) \]
\[ = \log \left(\frac{180}{5}\right) = \log 36 \]

6. **Now simplify the right-hand side:**
\[ 2(\log a - \log 5) = 2 \log \left(\frac{a}{5}\right) = \log \left(\left(\frac{a}{5}\right)^2\right) = \log \left(\frac{a^2}{25}\right) \]

7. **Set the left-hand side equal to the right-hand side:**
\[ \log 36 = \log \left(\frac{a^2}{25}\right) \]

8. **Since the logarithms are equal, we can set the arguments equal:**
\[ 36 = \frac{a^2}{25} \]

9. **Multiply both sides by 25:**
\[ 36 \cdot 25 = a^2 \]
\[ a^2 = 900 \]

10. **Take the square root of both sides:**
\[ a = 30 \quad (\text{since } a > 0) \]

### Final Answer:
\[ a = 30 \]