Basics & LogarithmseasyFree

Basics & Logarithms — JEE Maths practice question

JEE Maths question with a full step-by-step solution.

Question
Sum of all the values of xx satisfying the equation log⁡17log⁡11(x+11+x)=0\log_{17}\log_{11}(\sqrt{x+11}+\sqrt{x}) = 0 is:
A2525correct
B3636
C171171
D00
Solution
Step 1 Remove the loglog
log⁡11(x+11+x)=170=1  ⟹  x+11+x=11\log_{11}(\sqrt{x+11}+\sqrt{x}) = 17^0 = 1 \implies \sqrt{x+11}+\sqrt{x} = 11
 Step 2: Solve
x+11=11−x  ⟹  x+11=121−22x+x\sqrt{x+11} = 11 - \sqrt{x} \implies x+11 = 121 - 22\sqrt{x} + x
22x=110  ⟹  x=5  ⟹  x=2522\sqrt{x} = 110 \implies \sqrt{x} = 5 \implies x = 25
 This satisfies the original equation, so the sum of all values is 25.
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