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 numerator and the denominator separately.

### Step 1: Analyze the Numerator
The numerator is

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

This implies that

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

### Step 2: Solve for \(\log_{4}x\)
From the previous step, we can rewrite it in exponential form:

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

### Step 3: Solve for \(x\)
Now, we convert this logarithmic equation back to exponential form:

\[ x = 4^8. \]

Calculating \(4^8\):

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

### Step 4: Analyze the Denominator
The denominator is

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

We need to ensure that this expression is not equal to zero.

### Step 5: Substitute \(x\) into the Denominator
Substituting \(x = 2^{16}\):

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

Now we need to find \(\log_{4}(16)\):

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

### Step 6: Substitute into the Denominator
Now we have:

\[ \log_{5}(\log_{4}(\log_{y}(16))). \]

### Step 7: Set the Condition for the Denominator
For the denominator to be defined and not equal to zero, we need:

\[ \log_{4}(\log_{y}(16)) \neq 0. \]

This means:

\[ \log_{y}(16) \neq 4^0 = 1. \]

### Step 8: Solve for \(y\)
From \(\log_{y}(16) = 1\), we get:

\[ y^1 = 16 \implies y = 16. \]

### Step 9: Determine the Limits for \(y\)
We also have the condition \(y \neq 1\) and \(y\) must be between \(c\) and \(b\) where \(b\) is as large as possible. The smallest value \(c\) can take is \(2\) (since \(\log_{4}(2) > 0\)).

### Step 10: Identify Values of \(a\), \(b\), and \(c\)
From our analysis:
- \(a = 1\) (since \(y \neq 1\)),
- \(b = 16\) (the largest value for \(y\)),
- \(c = 2\) (the smallest value for \(y\)).

### Step 11: Calculate \(a + b + c\)
Now we can calculate:

\[ a + b + c = 1 + 16 + 2 = 19. \]

### Final Answer
Thus, the value of \(a + b + c\) is

\[ \boxed{19}. \]