CA Foundation · Quantitative Aptitude · Ratio and Proportion, Indices and Logarithms
If log₃ (x − 2) + log₃ (x + 6) = 2, then the value of x is:
The value of x is 3. Combining the logs gives (x−2)(x+6)=9, so x²+4x−21=0, giving x=3 or x=-7. Logarithms need positive arguments, so x=-7 is rejected, leaving only x=3.
- A3Correct
- B-7
- C3 or -7
- D5
Explanation
Product law: log₃ [(x−2)(x+6)] = 2, so (x−2)(x+6) = 9, giving x² + 4x − 12 = 9, i.e., x² + 4x − 21 = 0, so (x+7)(x−3) = 0. x = 3 or -7. For x = -7, x−2 is negative and the logarithm is undefined, so only x = 3 is valid. Check: log₃ 1 + log₃ 9 = 0 + 2 = 2.
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