Question

Each of the following questions is followed by two statements, I and II. Answer as follows:
Mark option (1) if the question can be answered using statement I alone, but not using statement II alone.
Mark option (2) if the question can be answered using statement II alone, but not using statement I alone.
Mark option (3) if the question can be answered using both statements I and II together, but not by using either statement alone.
Mark option (4) if the question cannot be answered even using both statements I and II together.

What is the value of \(x\)?
I. \( \log_2 2^x = x \)
II. \( \log_3 x = 0 \)

• A. The question can be answered using statement I alone, but not using statement II alone.
• B. The question can be answered using statement II alone, but not using statement I alone.
• C. The question can be answered using both statements I and II together, but not by using either statement alone.
• D. The question cannot be answered even using both statements I and II together.

Hint:

Remember that \( \log_a a^k = k \) is always true for any \(x\), so it can never pin down a single value; check whether a statement is a fixed identity before using it as new information.

Answer 1

The Correct Option is B

Step 1: Read statement I.
Statement I says \( \log_2 2^x = x \).
By the basic rule of logarithms, \( \log_a a^k = k \) for any base \(a\) and any real number \(k\).
So \( \log_2 2^x = x \) is true for every value of \(x\), not just one particular value.
It is an identity, so it does not fix \(x\) to a single number. Statement I alone cannot tell us the value of \(x\).

Step 2: Read statement II.
Statement II says \( \log_3 x = 0 \).
Convert this to exponential form: \( x = 3^0 \).
Since any nonzero number raised to the power \(0\) is \(1\), we get \( x = 1 \).
This gives one exact value of \(x\), so statement II alone is enough to answer the question.

Step 3: Combine the findings.
Statement I alone gives no unique value of \(x\) since it holds for all \(x\).
Statement II alone gives the unique value \( x = 1 \).
So the question can be answered using statement II alone, but not using statement I alone.

Final Answer:
Statement II alone is sufficient, statement I alone is not.
\[ \boxed{\text{Option (2)}} \]