Calculate: log_{2}(3x+1)-3log_{4}(x^2)+2log_{2}(x)

Expression: $\log_{ 2 }({ 3x+1 })-3\log_{ 4 }({ {x}^{2} })+2\log_{ 2 }({ x })$

Write the number in exponential form with the base of $2$

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

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

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