Chapter 6.5, Problem 17E

To determine

To calculate: The expanded form of the expression $\ln \frac{x}{3y}$ .

Answer to Problem 17E

The required expanded form of the expression $\ln \frac{x}{3y}$ is $\ln x-\text{ln}3-\ln y$ .

Explanation of Solution

Given information:

The given expression is $\ln \frac{x}{3y}$ .

Formula used:

Quotient property of logarithm:

  ${\log }_b\frac{m}{n}={\log }_{b}m-{\log }_{b}n$

Product property of logarithm:

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

Calculation:

Consider the expression $\ln \frac{x}{3y}$ .

Expand the expression using quotient property of logarithm ${\log }_b\frac{m}{n}={\log }_{b}m-{\log }_{b}n$ .

  $\ln \frac{x}{3y}=\ln x-\ln 3y$

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

  $\begin{array}{c}\ln \frac{x}{3y}=\ln x-\ln 3y\\ =\ln x-\left(\ln 3+\ln y\right)\\ =\ln x-\text{ln}3-\ln y\end{array}$

Hence, the required expanded form of the expression $\ln 3x^4$ is $\ln x-\text{ln}3-\ln y$ .