The value of `N` satisfying `log_(a)[1+log_(b){1+log_(c)(1+log_(p)N)}]=0` is

Options

A. `2`
B. `0`
C. `5`
D. `1`

Explanation

To solve the equation \( \log_a \left( 1 + \log_b \left( 1 + \log_c \left( 1 + \log_p N \right) \right) \right) = 0 \), we will proceed step by step.

### Step 1: Simplify the logarithmic equation
Starting with the equation:
\[ \log_a \left( 1 + \log_b \left( 1 + \log_c \left( 1 + \log_p N \right) \right) \right) = 0 \]
Using the property of logarithms, we know that if \( \log_a x = 0 \), then \( x = 1 \). Therefore, we can rewrite the equation as:
\[ 1 + \log_b \left( 1 + \log_c \left( 1 + \log_p N \right) \right) = 1 \]

### Step 2: Isolate the logarithm
Subtracting 1 from both sides gives:
\[ \log_b \left( 1 + \log_c \left( 1 + \log_p N \right) \right) = 0 \]
Again, applying the logarithmic property, we have:
\[ 1 + \log_c \left( 1 + \log_p N \right) = 1 \]

### Step 3: Continue isolating the logarithm
Subtracting 1 from both sides again results in:
\[ \log_c \left( 1 + \log_p N \right) = 0 \]
This implies:
\[ 1 + \log_p N = 1 \]

### Step 4: Isolate the innermost logarithm
Subtracting 1 from both sides gives:
\[ \log_p N = 0 \]
Using the logarithmic property once more, we conclude:
\[ N = p^0 \]

### Step 5: Final value of N
Since \( p^0 = 1 \), we find that:
\[ N = 1 \]

Thus, the value of \( N \) satisfying the original equation is:
\[ \boxed{1} \]