Number of solution(s) satisfying the equations, $3 x^{2} - 2 x^{3} = \log_{2} (x^{2} + 1) - \log_{2} x$ is

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$3$

$2$

$1$

None of these

answer is C.

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Detailed Solution

Given equation is $3 x^{2} - 2 x^{3} = \log_{2} (x^{2} + 1) - \log_{2} x$ .

It is required to find the number of solutions which satisfies the given equation.

Using the basic logarithmic property, $\log m - \log n = \log (\frac{m}{n})$

$3 x^{2} - 2 x^{3} = \log_{2} (x^{2} + 1) - \log_{2} x 3 x^{2} - 2 x^{3} = \log_{2} (\frac{x^{2} + 1}{x})$

Writing the equation in exponential form

$2^{3 x^{2} - 2 x^{3}} = \frac{x^{2} + 1}{x}$

$x$ is not defined when $x = 0, 2, 3$

Only $x = 1$ satisfies this equation.

Hence, number of solutions for the given equation is $1$ .

Therefore, the correct answer is option $(3)$ .

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