IF `10^(logp{logq(logr^x)})`=1 and `log _q{log_r(log_px)}=0`.
The value of x is

Options

A. `q^r`
B. `r^q`
C. `r^p`
D. rq

Answer

To solve the given equations, we will work through them step by step.

### 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: Solve for \(\log_q(\log_r(x))\)
Using the property of logarithms, we can rewrite the equation:
\[ \log_q(\log_r(x)) = p^0 = 1 \]

### Step 3: Solve for \(\log_r(x)\)
Again, applying the property of logarithms:
\[ \log_r(x) = q^1 = q \]

### Step 4: Solve for \(x\)
Now, we can express \(x\) in terms of \(r\):
\[ x = r^q \]

### Step 5: Analyze the second equation
Now we move to the second equation:
\[ \log_q(\log_r(\log_p(x))) = 0 \]
Similar to before, we equate the exponent to zero:
\[ \log_r(\log_p(x)) = q^0 = 1 \]

### Step 6: Solve for \(\log_p(x)\)
Using the property of logarithms:
\[ \log_p(x) = r^1 = r \]

### Step 7: Solve for \(x\)
Now, we can express \(x\) in terms of \(p\):
\[ x = p^r \]

### Summary of Results
From the two equations, we derived:
1. \( x = r^q \)
2. \( x = p^r \)

### Final Answer
Thus, the value of \(x\) can be expressed in two forms:
- \( x = r^q \)
- \( x = p^r \)