If a= ` log 2 0 log 3 , b = log 3 - log 5 and c= log 2.5` find the value of :
a + b+ c

Answer

To solve the problem, we need to find the value of \( a + b + c \) where:

- \( a = \log 20 \cdot \log 3 \)
- \( b = \log 3 - \log 5 \)
- \( c = \log 2.5 \)

Let's break it down step by step.

### Step 1: Simplifying \( a \)
We know that:
\[ a = \log 20 \cdot \log 3 \]
We can express \( \log 20 \) as:
\[ \log 20 = \log (4 \cdot 5) = \log 4 + \log 5 \]
And since \( \log 4 = \log (2^2) = 2 \log 2 \), we have:
\[ \log 20 = 2 \log 2 + \log 5 \]
Thus,
\[ a = (2 \log 2 + \log 5) \cdot \log 3 \]

### Step 2: Simplifying \( b \)
Using the properties of logarithms, we can simplify \( b \):
\[ b = \log 3 - \log 5 = \log \left( \frac{3}{5} \right) \]

### Step 3: Simplifying \( c \)
We can rewrite \( c \):
\[ c = \log 2.5 = \log \left( \frac{25}{10} \right) = \log 25 - \log 10 \]
Since \( \log 25 = \log (5^2) = 2 \log 5 \) and \( \log 10 = \log (2 \cdot 5) = \log 2 + \log 5 \), we have:
\[ c = 2 \log 5 - (\log 2 + \log 5) = 2 \log 5 - \log 2 - \log 5 = \log 5 - \log 2 \]

### Step 4: Combining \( a \), \( b \), and \( c \)
Now we can combine all three:
\[ a + b + c = (2 \log 2 + \log 5) \cdot \log 3 + \log \left( \frac{3}{5} \right) + (\log 5 - \log 2) \]

### Step 5: Expanding and simplifying
Expanding \( a \):
\[ = 2 \log 2 \cdot \log 3 + \log 5 \cdot \log 3 + \log 3 - \log 5 + \log 5 - \log 2 \]
The \( \log 5 \) terms cancel out:
\[ = 2 \log 2 \cdot \log 3 + \log 3 - \log 2 \]

### Step 6: Final simplification
Now we can factor out \( \log 3 \):
\[ = \log 3 (2 \log 2 + 1) - \log 2 \]

This is the final expression for \( a + b + c \).

### Final Answer
The value of \( a + b + c \) is:
\[ \log 3 (2 \log 2 + 1) - \log 2 \]