Let `3^(a) = 4, 4^(b) = 5, 5^(c) = 6, 6^(d) = 7, 7^(e) = 8` and `8^(f) = 9`. The value of the product (abcdef), is

Options

A. 1
B. 2
C. `sqrt(6)`
D. 3

Answer

To solve the problem, we need to find the product \( abcdef \) given the equations:

1. \( 3^a = 4 \)
2. \( 4^b = 5 \)
3. \( 5^c = 6 \)
4. \( 6^d = 7 \)
5. \( 7^e = 8 \)
6. \( 8^f = 9 \)

We can express each variable in terms of logarithms:

### Step 1: Express \( a \)
From the equation \( 3^a = 4 \), we can take the logarithm of both sides:
\[ a = \log_3(4) = \frac{\log(4)}{\log(3)} \]

### Step 2: Express \( b \)
From the equation \( 4^b = 5 \):
\[ b = \log_4(5) = \frac{\log(5)}{\log(4)} \]

### Step 3: Express \( c \)
From the equation \( 5^c = 6 \):
\[ c = \log_5(6) = \frac{\log(6)}{\log(5)} \]

### Step 4: Express \( d \)
From the equation \( 6^d = 7 \):
\[ d = \log_6(7) = \frac{\log(7)}{\log(6)} \]

### Step 5: Express \( e \)
From the equation \( 7^e = 8 \):
\[ e = \log_7(8) = \frac{\log(8)}{\log(7)} \]

### Step 6: Express \( f \)
From the equation \( 8^f = 9 \):
\[ f = \log_8(9) = \frac{\log(9)}{\log(8)} \]

### Step 7: Calculate the product \( abcdef \)
Now, we can find the product \( abcdef \):
\[ abcdef = \left(\frac{\log(4)}{\log(3)}\right) \left(\frac{\log(5)}{\log(4)}\right) \left(\frac{\log(6)}{\log(5)}\right) \left(\frac{\log(7)}{\log(6)}\right) \left(\frac{\log(8)}{\log(7)}\right) \left(\frac{\log(9)}{\log(8)}\right) \]

Notice that in this product, all the logarithms will cancel out:
\[ abcdef = \frac{\log(9)}{\log(3)} \]

### Step 8: Simplify \( \frac{\log(9)}{\log(3)} \)
We know that \( \log(9) = \log(3^2) = 2 \log(3) \):
\[ abcdef = \frac{2 \log(3)}{\log(3)} = 2 \]

### Final Answer
Thus, the value of the product \( abcdef \) is:
\[ \boxed{2} \]