$\log_{ 2 }({ 3x+1 })-3\log_{ {2}^{2} }({ {x}^{2} })+2\log_{ 2 }({ x })$
Use $x \times \log_{ a }({ b })=\log_{ a }({ {b}^{x} })$ to transform the expression$\log_{ 2 }({ 3x+1 })-3\log_{ {2}^{2} }({ {x}^{2} })+\log_{ 2 }({ {x}^{2} })$
Use $\log_{ {a}^{y} }({ b })=\frac{ 1 }{ y } \times \log_{ a }({ b })$ to transform the expression$\log_{ 2 }({ 3x+1 })-3 \times \frac{ 1 }{ 2 } \times \log_{ 2 }({ {x}^{2} })+\log_{ 2 }({ {x}^{2} })$
Calculate the product$\log_{ 2 }({ 3x+1 })-\frac{ 3 }{ 2 } \times \log_{ 2 }({ {x}^{2} })+\log_{ 2 }({ {x}^{2} })$
Use $x \times \log_{ a }({ b })=\log_{ a }({ {b}^{x} })$ to transform the expression$\log_{ 2 }({ 3x+1 })+\log_{ 2 }({ {\left( {x}^{2} \right)}^{-\frac{ 3 }{ 2 }} })+\log_{ 2 }({ {x}^{2} })$
Use the logarithmic product and quotient rules to simplify the expression$\log_{ 2 }({ \left( 3x+1 \right) \times {\left( {x}^{2} \right)}^{-\frac{ 3 }{ 2 }} \times {x}^{2} })$
Simplify the expression by multiplying exponents$\log_{ 2 }({ \left( 3x+1 \right){x}^{-3} \times {x}^{2} })$
Calculate the product$\log_{ 2 }({ \left( 3x+1 \right){x}^{-1} })$
Distribute ${x}^{-1}$ through the parentheses$\log_{ 2 }({ 3+{x}^{-1} })$
Any expression raised to the power of $-1$ equals its reciprocal$\log_{ 2 }({ 3+\frac{ 1 }{ x } })$
Write all numerators above the common denominator$\log_{ 2 }({ \frac{ 3x+1 }{ x } })$