Chapter 6.5, Problem 30E

To determine

To calculate: The condensed form of the expression ${\log }_{3}4+2{\log }_3\frac{1}{2}+{\log }_{7}x$ .

Answer to Problem 30E

The required condensed form of the expression ${\log }_{3}4+2{\log }_3\frac{1}{2}+{\log }_{7}x$ is ${\log }_{3}x$ .

Explanation of Solution

Given information:

The given expression is ${\log }_{3}4+2{\log }_3\frac{1}{2}+{\log }_{3}x$ .

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:

Consider the given expression as $\ln 12-\ln 4$ .

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

  $\begin{array}{c}{\log }_{3}4+2{\log }_3\frac{1}{2}+{\log }_{3}x={\log }_{3}4+{\log }_3{\left(\frac{1}{2}\right)}^2+{\log }_{3}x\\ ={\log }_{3}4+{\log }_3\frac{1}{4}+{\log }_{3}x\end{array}$

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

  $\begin{array}{c}{\log }_{3}4+2{\log }_3\frac{1}{2}+{\log }_{7}x={\log }_{3}4+{\log }_3{\left(\frac{1}{2}\right)}^2+{\log }_{7}x\\ ={\log }_{3}4\cdot \frac{1}{4}\cdot x\\ ={\log }_{3}x\end{array}$

Hence, the required condensed form of the expression ${\log }_{3}4+2{\log }_3\frac{1}{2}+{\log }_{7}x$ is ${\log }_{3}x$ .