If `log_2 (log_2 (log_3 x)) = log_2 (log_3 (log_2 y))=0`, then the value of `(x+y)` is

Options

A. 17
B. 9
C. 21
D. 19

Answer

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

1. \( \log_2 (\log_2 (\log_3 x)) = 0 \)
2. \( \log_2 (\log_3 (\log_2 y)) = 0 \)

### Step 1: Solve for \( x \)

From the first equation, we have:

\[ \log_2 (\log_2 (\log_3 x)) = 0 \]

Using the property of logarithms that states \( \log_b a = c \) implies \( a = b^c \), we can rewrite this as:

\[ \log_2 (\log_3 x) = 2^0 = 1 \]

Now, applying the logarithm property again:

\[ \log_3 x = 2^1 = 2 \]

Using the property again:

\[ x = 3^2 = 9 \]

### Step 2: Solve for \( y \)

Now, we move to the second equation:

\[ \log_2 (\log_3 (\log_2 y)) = 0 \]

Again, applying the logarithm property:

\[ \log_3 (\log_2 y) = 2^0 = 1 \]

Now applying the logarithm property again:

\[ \log_2 y = 3^1 = 3 \]

Using the property again:

\[ y = 2^3 = 8 \]

### Step 3: Calculate \( x + y \)

Now that we have both \( x \) and \( y \):

\[ x + y = 9 + 8 = 17 \]

Thus, the final answer is:

\[ \boxed{17} \]