Which one of the following is correct ?
`(i) 13^(31) > 31^13 " " (ii) 10^(100) < 100^(10)`
`(iii) 2^(32) < 32^2`.

Options

A. (i) and (ii)
B. (i) and (iii)
C. (ii) and (iii)
D. (i) only

Answer

To solve the question, we need to evaluate each of the three statements given:

1. **Statement (i):** \( 13^{31} > 31^{13} \)

To compare \( 13^{31} \) and \( 31^{13} \), we can take the logarithm of both sides.

\[ \log(13^{31}) = 31 \cdot \log(13) \]
\[ \log(31^{13}) = 13 \cdot \log(31) \]

Now we need to compare \( 31 \cdot \log(13) \) and \( 13 \cdot \log(31) \).

Dividing both sides by \( 13 \cdot 31 \) gives us:

\[ \frac{\log(13)}{13} \quad \text{and} \quad \frac{\log(31)}{31} \]

We can evaluate these logarithmic values:

- \( \log(13) \approx 1.113943 \)
- \( \log(31) \approx 1.491654 \)

Now calculating:

\[ \frac{1.113943}{13} \approx 0.0857 \]
\[ \frac{1.491654}{31} \approx 0.0488 \]

Since \( 0.0857 > 0.0488 \), we conclude that:

\[ 31 \cdot \log(13) > 13 \cdot \log(31) \implies 13^{31} > 31^{13} \]

Thus, **Statement (i) is true.**

2. **Statement (ii):** \( 10^{100} < 100^{10} \)

We can rewrite \( 100^{10} \) as \( (10^2)^{10} = 10^{20} \).

Now we compare:

\[ 10^{100} < 10^{20} \]

This is clearly false since \( 100 > 20 \). Therefore, **Statement (ii) is false.**

3. **Statement (iii):** \( 2^{32} < 32^{2} \)

We can rewrite \( 32 \) as \( 2^5 \), so:

\[ 32^2 = (2^5)^2 = 2^{10} \]

Now we compare:

\[ 2^{32} < 2^{10} \]

This is also false since \( 32 > 10 \). Therefore, **Statement (iii) is false.**

### Conclusion:
The only correct statement is (i): \( 13^{31} > 31^{13} \).

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