Question

Use the Properties of Logarithms to expand the logarithm ${\log }_2\left(5x^4{y}^2\right)$. Simplify, if possible

Short answer

The expanded form of the logarithm${\log }_2\left(5x^4{y}^2\right)$is${\log }_2\left(5x^4{y}^2\right)={\log }_2\left(5\right)+4{\log }_2\left(x\right)+2{\log }_2\left(y\right)$

Step-by-step solution

Step 1. given data   

Given logarithm expressions is${\log }_2\left(5x^4{y}^2\right)$

Step 2. Apply product Properties of Logarithms 

Use product Properties of Logarithms ${\log }_a\left(m·n\right)={\log }_{a}m+{\log }_{a}n$

sorole="math" localid="1644956884292" ${\log }_2\left(5x^4{y}^2\right)={\log }_2\left(5\right)+{\log }_2\left(x^4\right)+{\log }_2\left(y^2\right)$

Step 3. Apply power Property of the logarithm 

Use power Property of the logarithm ${\log }_a\left(m^p\right)=p{\log }_a\left(m\right)$

so$$\begin{gathered} {\log }_2\left(5x^4{y}^2\right)={\log }_2\left(5\right)+{\log }_2\left(x^4\right)+{\log }_2\left(y^2\right) \\ {\log }_2\left(5x^4{y}^2\right)={\log }_2\left(5\right)+4{\log }_2\left(x\right)+2{\log }_2\left(y\right) \end{gathered}$$