STATEMENT-1 : Number of solution of ` log |x| = theta^(x)` is two and
STATEMENT-2 : If ` log_(30) 3 - a , log_(30) 5 = b "then" log_(30) 8 = 3 (1 - a - b)` .

Options

A. Statemant-1 is True , Statement-2 is True, Statement -2 is a correct explanation for Statement-1
B. Statemant-1 is True , Statement-2 is True, Statement -2 is NOT a correct explanation for Statement-1
C. Statement-1 is True, Stetement-2 is False.
D. Statement-1 is False, Statement-2 is True

Answer

To solve the problem, we will analyze both statements one by one.

### Statement 1:
We need to determine the number of solutions for the equation:

\[ \log |x| = \theta^x \]

where \(\theta\) is a constant.

1. **Understanding the Graphs**:
- The function \(y = \log |x|\) has a vertical asymptote at \(x = 0\) and approaches \(-\infty\) as \(x\) approaches \(0\) from either side. It increases without bound as \(x\) approaches \(\infty\) and decreases without bound as \(x\) approaches \(-\infty\).
- The function \(y = \theta^x\) is an exponential function. If \(\theta > 1\), it increases from \(0\) to \(\infty\) as \(x\) increases. If \(0 < \theta < 1\), it decreases from \(\infty\) to \(0\).

2. **Finding Intersections**:
- For \(\theta > 1\): The graph of \(y = \theta^x\) will intersect the graph of \(y = \log |x|\) at most twice (once for \(x > 0\) and once for \(x < 0\)).
- For \(0 < \theta < 1\): The graph of \(y = \theta^x\) will intersect the graph of \(y = \log |x|\) at most once.

3. **Conclusion for Statement 1**:
- The statement claims that the number of solutions is two. This is only true when \(\theta > 1\). Therefore, the statement is **false** as it does not hold for all values of \(\theta\).

### Statement 2:
We need to verify the statement:

\[ \log_{30} 8 = 3(1 - a - b) \]

where \(a = \log_{30} 3\) and \(b = \log_{30} 5\).

1. **Expressing 8 in Terms of Logarithms**:
- We know that \(8 = 2^3\). Thus, we can express \(\log_{30} 8\) as:
\[ \log_{30} 8 = \log_{30} (2^3) = 3 \log_{30} 2 \]

2. **Finding \(\log_{30} 2\)**:
- We can express \(2\) in terms of \(3\) and \(5\) using the change of base formula:
\[ \log_{30} 2 = \log_{30} \left(\frac{30}{3 \cdot 5}\right) = \log_{30} 30 - \log_{30} 3 - \log_{30} 5 \]
- Since \(\log_{30} 30 = 1\), we have:
\[ \log_{30} 2 = 1 - a - b \]

3. **Substituting Back**:
- Now substituting this back into our expression for \(\log_{30} 8\):
\[ \log_{30} 8 = 3(1 - a - b) \]

4. **Conclusion for Statement 2**:
- This confirms that the statement is **true**.

### Final Conclusion:
- Statement 1 is **false**.
- Statement 2 is **true**.

Thus, the answer is that the first statement is false and the second statement is true.