Chapter 6.5, Problem 20E

To determine

To calculate: The expanded form of the expression ${\log }_5\sqrt[3]{x^{2}y}$

Answer to Problem 20E

The required expanded form of the expression ${\log }_5\sqrt[3]{x^{2}y}$ is $\frac{2}{3}{\log }_{5}x+\frac{1}{3}{\log }_{5}y$ .

Explanation of Solution

Given information:

The expression = ${\log }_5\sqrt[3]{x^{2}y}$

Formula used:

Power property of logarithm:

  ${\log }_b{m}^n=n{\log }_{b}m$

Product property of logarithm:

  ${\log }_{b}mn={\log }_{b}m+{\log }_{b}n$

Calculation:

The given expression is ${\log }_5\sqrt[3]{x^{2}y}$ .

Expand the expression using power property of logarithm, ${\log }_b{m}^n=n{\log }_{b}m$ .

  $\begin{array}{c}{\log }_5\sqrt[3]{x^{2}y}={\log }_5{\left(x^{2}y\right)}^{\frac{1}{3}}\\ =\frac{1}{3}{\log }_5{x}^{2}y\end{array}$

Expand the above expression using product property of logarithm, ${\log }_{b}mn={\log }_{b}m+{\log }_{b}n$ .

  $\begin{array}{c}{\log }_5\sqrt[3]{x^{2}y}=\frac{1}{3}{\log }_5{x}^{2}y\\ =\frac{1}{3}{\log }_5{x}^2+\frac{1}{3}{\log }_{5}y\end{array}$

Expand the expression using power property of logarithm, ${\log }_b{m}^n=n{\log }_{b}m$ .

  $\begin{array}{c}{\log }_5\sqrt[3]{x^{2}y}=\frac{1}{3}{\log }_5{x}^2+\frac{1}{3}{\log }_{5}y\\ =\frac{2}{3}{\log }_{5}x+\frac{1}{3}{\log }_{5}y\end{array}$

Hence, the required expanded form of the expression ${\log }_5\sqrt[3]{x^{2}y}$ is $\frac{2}{3}{\log }_{5}x+\frac{1}{3}{\log }_{5}y$ .