Number of solution(s) satisfying the equations, $3x^2-2x^3={\log }_2\left(x^2+1\right)-{\log }_{2}x$ is

Given equation is $3x^2-2x^3={\log }_2\left(x^2+1\right)-{\log }_{2}x$.

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

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

$$\begin{gathered} 3x^2-2x^3={\log }_2\left(x^2+1\right)-{\log }_{2}x \\ 3x^2-2x^3={\log }_2\left(\frac{x^2+1}{x}\right) \end{gathered}$$

Writing the equation in exponential form

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

$x$ 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 $\left(3\right)$.