Question

Use the Properties of Logarithms to expand the logarithm${\log }_3\left(7x^5{y}^3\right)$. Simplify, if possible.

Short answer

The expanded form of the logarithm${\log }_3\left(7x^5{y}^3\right)$ is${\log }_3\left(7x^5{y}^3\right)={\log }_{3}7+5{\log }_{3}x+3{\log }_{3}y$

Step-by-step solution

Step 1. given data   

Given logarithm expressions is${\log }_3\left(7x^5{y}^3\right)$

Step 2. Apply product Properties of Logarithms 

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

solocalid="1644957182189" ${\log }_3\left(7x^5{y}^3\right)={\log }_3\left(7\right)+{\log }_3\left(x^5\right)+{\log }_3\left(y^3\right)$

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$$\begin{gathered} {\log }_3\left(7x^5{y}^3\right)={\log }_3\left(7\right)+{\log }_3\left(x^5\right)+{\log }_3\left(y^3\right) \\ {\log }_3\left(7x^5{y}^3\right)={\log }_{3}7+5{\log }_{3}x+3{\log }_{3}y \end{gathered}$$