Chapter 6.5, Problem 29E

To determine

To calculate: The condensed form of the expression $5\ln 2+7\ln x+4\ln y$

Answer to Problem 29E

The required condensed form of the expression $5\ln 2+7\ln x+4\ln y$ is $\ln 32x^7{y}^4$ .

Explanation of Solution

Given information:

The expression = $5\ln 2+7\ln x+4\ln y$

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 $\ln 12-\ln 4$ .

Use the power property to condense the above expression, ${\log }_b{m}^n=n{\log }_{b}m$ .

  $5\ln 2+7\ln x+4\ln y=\ln 2^5+\ln x^7+\ln y^4$

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

  $\begin{array}{c}5\ln 2+7\ln x+4\ln y=\ln 2^5+\ln x^7+\ln y^4\\ =\ln 32+\ln x^7+\ln y^4\\ =\ln 32x^7{y}^4\end{array}$

Hence, the required condensed form of the expression $5\ln 2+7\ln x+4\ln y$ is $\ln 32x^7{y}^4$ .