`10^(log_p(log_q(log_r(x))))=1` and `log_q(log_r(log_p(x)))=0`, then 'p' is equals a. `r^(q/r)` b. `rq` c. 1 d. `r^(r/q)`

Options

A. `log _q{log_r(log_px)}=0`.
B. rq
C. 1
D. `r^(r//p)`

Answer

To solve the equations given in the problem, we will follow a systematic approach using logarithmic properties.

### Given:
1. \( 10^{\log_p(\log_q(\log_r(x)))} = 1 \)
2. \( \log_q(\log_r(\log_p(x))) = 0 \)

### Step 1: Analyze the first equation
From the first equation, we know that:
\[ 10^{\log_p(\log_q(\log_r(x)))} = 1 \]

Since \( 10^0 = 1 \), we can equate the exponent to zero:
\[ \log_p(\log_q(\log_r(x))) = 0 \]

### Step 2: Convert the logarithmic equation
Using the property of logarithms, we know that:
\[ \log_p(a) = 0 \Rightarrow a = p^0 = 1 \]

Thus, we have:
\[ \log_q(\log_r(x)) = 1 \]

### Step 3: Analyze the second equation
Now, let's analyze the second equation:
\[ \log_q(\log_r(\log_p(x))) = 0 \]

Again, using the property of logarithms:
\[ \log_q(a) = 0 \Rightarrow a = q^0 = 1 \]

This gives us:
\[ \log_r(\log_p(x)) = 1 \]

### Step 4: Convert this logarithmic equation
Using the property of logarithms again:
\[ \log_r(b) = 1 \Rightarrow b = r^1 = r \]

Thus, we have:
\[ \log_p(x) = r \]

### Step 5: Solve for \( x \)
Using the property of logarithms:
\[ \log_p(x) = r \Rightarrow x = p^r \]

### Step 6: Substitute \( x \) back into the first equation
Now we substitute \( x = p^r \) into the equation derived from the first equation:
\[ \log_q(\log_r(x)) = 1 \]
Substituting \( x \):
\[ \log_q(\log_r(p^r)) = 1 \]

### Step 7: Calculate \( \log_r(p^r) \)
Using the property of logarithms:
\[ \log_r(p^r) = r \cdot \log_r(p) \]

Thus:
\[ \log_q(r \cdot \log_r(p)) = 1 \]

### Step 8: Solve for \( p \)
Now we know:
\[ \log_q(r \cdot \log_r(p)) = 1 \Rightarrow r \cdot \log_r(p) = q \]

From this, we can express \( \log_r(p) \):
\[ \log_r(p) = \frac{q}{r} \]

### Step 9: Convert back to \( p \)
Using the property of logarithms:
\[ p = r^{\frac{q}{r}} \]

### Conclusion
Thus, we find that:
\[ p = r^{\frac{q}{r}} \]

### Final Answer
The correct option is:
**a. \( r^{\frac{q}{r}} \)**