Which of the following statement(s) is/are true?
I. `2^(600)gt7^(900)`
II. `3^(400)gt9^(500)`

Options

A. a)Only I
B. b)Only II
C. c)Both I and II
D. d)Neither I nor II

Answer

To determine the truth of the statements given, we will analyze each statement one by one.

### Statement I: \(2^{600} > 7^{900}\)

1. **Convert the bases to a common logarithmic base**:
We can take the logarithm (base 10 or natural logarithm) of both sides to compare the two expressions.
\[ \log(2^{600}) > \log(7^{900}) \]

2. **Apply the logarithmic power rule**:
Using the property of logarithms that states \(\log(a^b) = b \cdot \log(a)\), we can rewrite the inequality:
\[ 600 \cdot \log(2) > 900 \cdot \log(7) \]

3. **Rearranging the inequality**:
We can divide both sides by 300 (which is positive, so the inequality remains the same):
\[ 2 \cdot \log(2) > 3 \cdot \log(7) \]

4. **Calculating the logarithm values**:
Using approximate values: \(\log(2) \approx 0.301\) and \(\log(7) \approx 0.845\):
\[ 2 \cdot 0.301 \approx 0.602 \]
\[ 3 \cdot 0.845 \approx 2.535 \]

5. **Comparing the results**:
Since \(0.602 < 2.535\), we conclude that:
\[ 2^{600} < 7^{900} \]
Therefore, Statement I is **false**.

### Statement II: \(3^{400} > 9^{500}\)

1. **Rewrite \(9\) in terms of \(3\)**:
We know that \(9 = 3^2\), so we can rewrite \(9^{500}\) as:
\[ 9^{500} = (3^2)^{500} = 3^{1000} \]

2. **Now compare the two expressions**:
We need to check if:
\[ 3^{400} > 3^{1000} \]

3. **Since the bases are the same, compare the exponents**:
If the bases are the same, we can compare the exponents directly:
\[ 400 > 1000 \]
This is clearly **false**.

Thus, Statement II is also **false**.

### Conclusion:
Both statements are false. Therefore, the correct answer is that neither statement is true.

### Final Answer:
Both statements I and II are false.

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