Question

Use the Properties of Logarithms to expand the logarithm ${\log }_4\sqrt[5]{\frac{x^4}{2y^3{z}^2}}$. Simplify, if possible

Short answer

The expanded form of the logarithm is${\log }_4\sqrt[5]{\frac{x^4}{2y^3{z}^2}}=\frac{1}{5}\left(4{\log }_{4}x-\frac{1}{2}-3{\log }_{4}y-2{\log }_{4}z\right)$

Step-by-step solution

Step 1. given data    

Given logarithm expressions is${\log }_4\sqrt[5]{\frac{x^4}{2y^3{z}^2}}$

Step 2. Apply power Property of the logarithm 

Rewrite the radical with a rational exponent

${\log }_4\sqrt[5]{\frac{x^4}{2y^3{z}^2}}={\log }_4{\left(\frac{x^4}{2y^3{z}^2}\right)}^{\frac{1}{5}}$

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

$$\begin{gathered} {\log }_4\sqrt[5]{\frac{x^4}{2y^3{z}^2}}={\log }_4{\left(\frac{x^4}{2y^3{z}^2}\right)}^{\frac{1}{5}} \\ {\log }_4\sqrt[5]{\frac{x^4}{2y^3{z}^2}}=\frac{1}{5}{\log }_4\left(\frac{x^4}{2y^3{z}^2}\right) \end{gathered}$$

Step 3. Apply the Quotient Property of logarithm 

Use the Quotient Property of logarithm ${\log }_a\left(\frac{m}{n}\right)={\log }_a\left(m\right)-{\log }_a\left(n\right)$

solocalid="1644958854832" $$\begin{gathered} {\log }_4\sqrt[5]{\frac{x^4}{2y^3{z}^2}}=\frac{1}{5}{\log }_4\left(\frac{x^4}{2y^3{z}^2}\right) \\ {\log }_4\sqrt[5]{\frac{x^4}{2y^3{z}^2}}=\frac{1}{5}\left({\log }_4{x}^4-{\log }_4\left(2y^3{z}^2\right)\right) \end{gathered}$$

Step 4. apply product Properties of Logarithms 

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

localid="1644959168978" $$\begin{gathered} {\log }_4\sqrt[5]{\frac{x^4}{2y^3{z}^2}}=\frac{1}{5}\left({\log }_4{x}^4-{\log }_4\left(2y^3{z}^2\right)\right) \\ {\log }_4\sqrt[5]{\frac{x^4}{2y^3{z}^2}}=\frac{1}{5}\left({\log }_4{x}^4-\left({\log }_{4}2+{\log }_4{y}^3+{\log }_4{z}^2\right)\right) \\ {\log }_4\sqrt[5]{\frac{x^4}{2y^3{z}^2}}=\frac{1}{5}\left({\log }_4{x}^4-{\log }_{4}2-{\log }_4{y}^3-{\log }_4{z}^2\right) \end{gathered}$$

Step 5. Apply power Property of the logarithm 

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

$$\begin{gathered} {\log }_4\sqrt[5]{\frac{x^4}{2y^3{z}^2}}=\frac{1}{5}\left({\log }_4{x}^4-{\log }_{4}2-{\log }_4{y}^3-{\log }_4{z}^2\right) \\ {\log }_4\sqrt[5]{\frac{x^4}{2y^3{z}^2}}=\frac{1}{5}\left(4{\log }_{4}x-\frac{1}{2}-3{\log }_{4}y-2{\log }_{4}z\right) \end{gathered}$$