Let `3^(a) = 4, 4^(b) = 5, 56(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 step by step, we start with the given 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 need to find the product \( abcdef \).

### Step 1: Express each variable in terms of logarithms

We can take logarithms on both sides of each equation to express \( a, b, c, d, e, \) and \( f \):

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

### Step 2: Write the product \( abcdef \)

Now we can write the product \( abcdef \):

\[ abcdef = \left(\frac{\log 4}{\log 3}\right) \left(\frac{\log 5}{\log 4}\right) \left(\frac{\log 6}{\log 5}\right) \left(\frac{\log 7}{\log 6}\right) \left(\frac{\log 8}{\log 7}\right) \left(\frac{\log 9}{\log 8}\right) \]

### Step 3: Simplify the product

Notice that in the product, many terms will cancel out:

\[ abcdef = \frac{\log 4}{\log 3} \cdot \frac{\log 5}{\log 4} \cdot \frac{\log 6}{\log 5} \cdot \frac{\log 7}{\log 6} \cdot \frac{\log 8}{\log 7} \cdot \frac{\log 9}{\log 8} \]

After cancellation, we are left with:

\[ abcdef = \frac{\log 9}{\log 3} \]

### Step 4: Simplify further

We can simplify \( \frac{\log 9}{\log 3} \):

Since \( 9 = 3^2 \), we have:

\[ \log 9 = \log(3^2) = 2 \log 3 \]

Thus,

\[ abcdef = \frac{2 \log 3}{\log 3} = 2 \]

### Final Answer

The value of the product \( abcdef \) is \( 2 \).

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