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: Treat statement I as an equation to solve.
Write $y = 2^x$. Statement I becomes $\log_2 y = x$, which is just $y = 2^x$ again.
So statement I simply restates $y = 2^x$ in a different form. It is true no matter what value $x$ takes: put $x = 0$, or $x = 1$, or $x = 5$, and the equation still checks out.
This means statement I places zero restriction on $x$. It cannot pin down a single value.


Step 2: Solve statement II directly.
Statement II gives $\log_3 x = 0$.
By the definition of a logarithm, $\log_3 x = 0$ means $3^0 = x$.
Since $3^0 = 1$, we get $x = 1$, and this is the only value of $x$ that works.


Step 3: Check what each statement tells us.
Statement I is an identity true for every real $x$, so it gives no clue about which particular $x$ the question wants.
Statement II by itself fixes $x$ uniquely as $1$, so we do not need statement I at all.


Step 4: Conclusion.
Statement II alone answers the question; statement I alone does not.
\[ \boxed{\text{Option (2): statement II alone is sufficient}} \]