If `a_(n) gt a_(n-1) gt …. gt a_(2) gt a_(1) gt 1`, then the value of `"log"_(a_(1))"log"_(a_(2)) "log"_(a_(3))…."log"_(a_(n))` is equal to

Options

A. 0
B. 1
C. 2
D. none of these

Explanation

To solve the problem, we start with the given condition:

1. **Given Condition**: \( a_n > a_{n-1} > \ldots > a_2 > a_1 > 1 \)

We need to find the value of:

\[ \log_{a_1}(\log_{a_2}(\log_{a_3}(\ldots(\log_{a_n}(x))\ldots))) \]

where \( x \) is some expression that we will determine.

### Step 1: Understanding the logarithmic expression

We can rewrite the expression as follows:

\[ \log_{a_1}(\log_{a_2}(\log_{a_3}(\ldots(\log_{a_n}(x))\ldots))) \]

### Step 2: Evaluating the innermost logarithm

Let's evaluate the innermost logarithm, \( \log_{a_n}(x) \). Since \( a_n > 1 \), we can choose \( x = a_n \):

\[ \log_{a_n}(a_n) = 1 \]

### Step 3: Moving to the next logarithm

Now, we substitute this back into the next logarithm:

\[ \log_{a_{n-1}}(1) \]

Since \( a_{n-1} > 1 \):

\[ \log_{a_{n-1}}(1) = 0 \]

### Step 4: Continuing the process

Now, we substitute this into the next logarithm:

\[ \log_{a_{n-2}}(0) \]

However, the logarithm of zero is undefined. Therefore, we need to be careful about the values we choose.

### Step 5: Generalizing the pattern

Continuing this pattern, we see that:

- \( \log_{a_n}(a_n) = 1 \)
- \( \log_{a_{n-1}}(1) = 0 \)
- \( \log_{a_{n-2}}(0) \) is undefined.

Thus, we can conclude that the entire expression evaluates to **undefined** due to the logarithm of zero.

### Final Conclusion

The value of \( \log_{a_1}(\log_{a_2}(\log_{a_3}(\ldots(\log_{a_n}(x))\ldots))) \) is undefined.

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