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Basics & Logarithms: Find Product Solutions Equation

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

Question
Find the product of all solutions of the equation
34log⁡3x−3log⁡27x=2log⁡3x\frac{3}{4\log_3 x} - 3\log_{27} x = 2\log_3 x
 is
Solution
Answer: 1
Step 1: Let t=log⁡3xt = \log_3 x.
34t−3⋅t3=2t  ⟹  34t−t=2t\frac{3}{4t} - 3 \cdot \frac{t}{3} = 2t \implies \frac{3}{4t} - t = 2t
 Step 2: Solve for tt:
34t=3t  ⟹  t2=14  ⟹  t=12 or t=−12\frac{3}{4t} = 3t \implies t^2 = \frac{1}{4} \implies t = \frac{1}{2} \text{ or } t = -\frac{1}{2}
 Step 3: Find xx values. t=12⇒x=31/2=3t = \dfrac{1}{2} \Rightarrow x = 3^{1/2} = \sqrt{3} t=−12⇒x=3−1/2=13t = -\dfrac{1}{2} \Rightarrow x = 3^{-1/2} = \dfrac{1}{\sqrt{3}} Both are positive, so both are valid. Step 4: Product of solutions =3⋅13=1= \sqrt{3} \cdot \dfrac{1}{\sqrt{3}} = 1. Answer: 1
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