If `log_a a. log_c a +log_c b. log_a b + log_a c. log_b c=3 `(where `a, b, c` are different positive real nu then find the value of `a bc.`

Explanation

To solve the equation \( \log_a a \cdot \log_c a + \log_c b \cdot \log_a b + \log_a c \cdot \log_b c = 3 \), we will use properties of logarithms and some algebraic manipulations.

### Step-by-Step Solution:

1. **Rewrite the logarithmic expressions:**
We can express the logarithms in terms of natural logarithms (or any common base). Using the change of base formula, we have:
\[
\log_a b = \frac{\log b}{\log a}, \quad \log_a c = \frac{\log c}{\log a}, \quad \log_b c = \frac{\log c}{\log b}
...