Basic Mathematics & Logarithm
Properties of Logarithms
Grade Class 11
Question:
<p>If \(\log_a x = b\) for permissible values of \(a\) and \(x\), identify the statement(s) which can be correct.</p>
If a and b are two irrational numbers then x can be rational.
If a is rational and b is irrational then x can be rational.
If a is irrational and b is rational then x can be rational.
If a is rational and b is rational then x can be rational.
Step-by-Step Solution
Key Concept: Test each statement using the identity $x=a^b$. Since the question asks which cases can occur, a single valid construction is enough.
Notice that the cleanest route is to simplify the structure before computing. A clever move here is to translate the logarithmic statement into a friendlier algebraic form. Use x = a^b and test each case by example. Each statement can be made true by a suitable choice. For example, irrational a and b can still give rational x; rational a with irrational b can give x = 2; irrational a with rational b can also give rational x; and rational a, rational b clearly can give rational x. Hence all four options are possible. Trap: This is an existence question, not a universal truth question; one valid example is enough. Now, we invoke the power of the relevant logarithmic identity, simplify carefully, and finally verify the domain so that no extraneous answer survives.
Correct Answer: A, B, C, D