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If logₓ(a) = b for permissible values of a and x, which of the following statements can be correct?

1. If a and b are both irrational, then x can be rational.
2. If a is rational and b is irrational, then x can be rational.
3. If a is irrational and b is rational, then x can be rational.
4. If a is rational and b is rational, then x can be rational.

Correct answer: If a and b are both irrational, then x can be rational.

Solution

x^b = a. For each case a rational x exists: A) x=2, b=sqrt(2), a=2^sqrt(2) (both irrational). B) x=2, b=log₂(3) (irrational), a=3 (rational). C) x=4, b=1/2 (rational), a=2 (irrational). D) x=2, b=3, a=8 (both rational). All four options are valid.

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