If the equation `(log_(12)(log_(8)(log_(4)x)))/(log_(5)(log_(4)(log_(y)(log_(2)x))))=0` has a solution for 'x' when `c lt y lt b, y ne a`, where 'b' is as large as possible, then the value of `(a+b+c)` is equals to :

Options

A. 18
B. 19
C. 20
D. 21

Answer

To solve the equation

\[ \frac{\log_{12}(\log_{8}(\log_{4}x))}{\log_{5}(\log_{4}(\log_{y}(\log_{2}x)))} = 0, \]

we need to analyze the conditions under which the equation holds true.

### Step 1: Understanding the Equation
The equation equals zero when the numerator is zero (since the denominator cannot be zero). Therefore, we need to solve:

\[ \log_{12}(\log_{8}(\log_{4}x)) = 0. \]

### Step 2: Solving the Numerator
The logarithm is zero when its argument is equal to 1. Hence, we have:

\[ \log_{8}(\log_{4}x) = 1. \]

### Step 3: Converting the Logarithm
This means:

\[ \log_{4}x = 8^1 = 8. \]

### Step 4: Solving for x
Now we convert this logarithm back to its exponential form:

\[ x = 4^8. \]

Calculating \(4^8\):

\[ 4^8 = (2^2)^8 = 2^{16} = 65536. \]

### Step 5: Analyzing the Denominator
Next, we need to ensure that the denominator is not zero:

\[ \log_{5}(\log_{4}(\log_{y}(\log_{2}x))) \neq 0. \]

This means:

\[ \log_{4}(\log_{y}(\log_{2}x)) \neq 1. \]

### Step 6: Solving the Denominator
The logarithm equals 1 when its argument equals 4:

\[ \log_{y}(\log_{2}x) = 4. \]

### Step 7: Converting the Logarithm
This implies:

\[ \log_{2}x = y^4. \]

### Step 8: Substituting for x
Substituting \(x = 65536\):

\[ \log_{2}(65536) = 16. \]

So we have:

\[ y^4 = 16. \]

### Step 9: Solving for y
Taking the fourth root:

\[ y = 2. \]

### Step 10: Finding the Range for y
The problem states that \(c < y < b\) and \(y \neq a\). We found \(y = 2\).

To find \(a\), \(b\), and \(c\):

- Since \(y = 2\), we can set \(c = 1\) (the next integer less than 2).
- For \(b\), we can set \(b = 16\) (the next integer greater than 2).
- We need to ensure \(y \neq a\). We can set \(a = 4\) (since \(y\) cannot be equal to \(a\)).

### Final Calculation
Now we can find \(a + b + c\):

\[ a + b + c = 4 + 16 + 1 = 21. \]

### Conclusion
Thus, the value of \(a + b + c\) is:

\[ \boxed{21}. \]