If `(log)_a x=b` for permissible values of `aa n dx ,` then identify the statement(s) which can be correct. If `aa n db` are two irrational numbers, then `x` can be rational. If `a` is rational and `b` is irrational, then `x` can be rational. If `a` is irrational and `b` is rational, then `x` can be rational. If `aa n d b` are rational, then `x` can be rational.

Options

A. If a and b are two irrational numbers, then x can be retional.
B. If a is rational and b is irrational, then x can be rational.
C. If a is irrational and b is rational, then x can be rational.
D. If a and b are rational, then x can be rational.

Explanation

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

\[ \log_a x = b \]

From this equation, we can express \(x\) in terms of \(a\) and \(b\):

\[ x = a^b \]

Now, we will analyze each of the four statements to determine whether they can be correct based on the values of \(a\) and \(b\).

### Statement 1:
**If \(a\) and \(b\) are two irrational numbers, then \(x\) can be rational.**

Let's take:
- \(a = \sqrt{2}^{\sqrt{2}}\) (which is irrational)
- \(b = \sqrt{2}\) (which is also irrational)

Now, we calculate \(x\):

\[ x = a^b = \left(\sqrt{2}^{\sqrt{2}}\right)^{\sqrt{2}} = \sqrt{2}^{(\sqrt{2} \cdot \sqrt{2})} = \sqrt{2}^2 = 2 \]

Since \(2\) is a rational number, this statement is **correct**.

### Statement 2:
**If \(a\) is rational and \(b\) is irrational, then \(x\) can be rational.**

Let’s take:
- \(a = 2\) (which is rational)
- \(b = \log_2 3\) (which is irrational)

Now, we calculate \(x\):

\[ x = a^b = 2^{\log_2 3} \]

Using the property of logarithms, we simplify:

\[ x = 3 \]

Since \(3\) is a rational number, this statement is also **correct**.

### Statement 3:
**If \(a\) is irrational and \(b\) is rational, then \(x\) can be rational.**

Let’s take:
- \(a = \sqrt{2}\) (which is irrational)
- \(b = 2\) (which is rational)

Now, we calculate \(x\):

\[ x = a^b = (\sqrt{2})^2 = 2 \]

Since \(2\) is a rational number, this statement is **correct**.

### Statement 4:
**If \(a\) and \(b\) are rational, then \(x\) can be rational.**

Let’s take:
- \(a = 2\) (which is rational)
- \(b = 3\) (which is rational)

Now, we calculate \(x\):

\[ x = a^b = 2^3 = 8 \]

Since \(8\) is a rational number, this statement is also **correct**.

### Conclusion:
All four statements can be correct based on the values of \(a\) and \(b\).

### Final Answer:
All four statements are correct.

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