First slide

Trigonometric Identities

Question

If $\log_{a} x = b$ for permissible values of a and x, then identify the statement(s) which can be correct.

Moderate

A

If a and b are two irrational numbers, then x can be rational.
B

If a is rational and b is irrational, then x can be rational.
C

If a is irrational and 6 is rational, then x can be rational.
D

If a and b are rational, then x can be rational.

Solution

$\log_{a} x = b or x = a^{b}$
$(1) For a = {\sqrt{2}}^{\sqrt{2}} \notin Q and b = \sqrt{2} \notin Q; x = {({\sqrt{2}}^{\sqrt{2}})}^{\sqrt{2}}$ which is rational.
$(2) For a = 2 \in Q and b = \log_{2} 3 \notin Q; x = 3$ which is rational.
$(3) For a = \sqrt{2} and b = 2; x = 2$
(4) The option is obviously correct.

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