If $$ 3^a = 4 $$, $$ 4^b = 5 $$, $$ 5^c = 6 $$, $$ 6^d = 7 $$, $$ 7^e = 8 $$, and $$ 8^f = 9 $$, then the value of the product $$ abcdef $$ is:

Question

Accepted Answer

1. **Express each exponent in terms of logarithms:**  
   - From $$ 3^a = 4 $$:  
     $$
     a = \frac{\log 4}{\log 3}
     $$  
   - Similarly:  
     $$
     b = \frac{\log 5}{\log 4}, \, c = \frac{\log 6}{\log 5}, \, d = \frac{\log 7}{\log 6}, \, e = \frac{\log 8}{\log 7}, \, f = \frac{\log 9}{\log 8}.
     $$

2. **Calculate the product $$ abcdef $$:**  
   - Substitute values:  
     $$
     abcdef = a \cdot b \cdot c \cdot d \cdot e \cdot f
     $$  
     $$
     abcdef = \frac{\log 4}{\log 3} \cdot \frac{\log 5}{\log 4} \cdot \frac{\log 6}{\log 5} \cdot \frac{\log 7}{\log 6} \cdot \frac{\log 8}{\log 7} \cdot \frac{\log 9}{\log 8}.
     $$

3. **Simplify the expression:**  
   - The intermediate terms cancel out:  
     $$
     abcdef = \frac{\log 9}{\log 3}.
     $$  
   - Using logarithmic properties ($$ \log a^b = b\log a $$):  
     $$
     abcdef = \frac{2\log 3}{\log 3} = 2.
     $$

Thus, the value of the product $$ abcdef $$ is **2**.