Property used:

Property 1:

\(\log_{2}(m)-\log_{2}(n)=\log_{2}(\frac{m}{n}) \)

Now to simplifying the given equation:

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

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

\(\log_{2}(\frac{3x-1}{x+1})=3\) [Using Property 1.]

Now taking antilog 2 and solving:

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

\((3x-1)=8(x+1)\)

\((3x-1)=8x+8\)

\(3x-8x=8x+1\)

\(-5x=9\)

\(x=-\frac{9}{5}\)

Since,

The solution does not satisfy the given equation.

Hence there is no solution for \(x \in R\)