Chapter 6.5, Problem 27E

To determine

To calculate: The condensed form of the expression ${\log }_{5}4+\frac{1}{3}{\log }_{5}x$

Answer to Problem 27E

The required condensed form of the expression ${\log }_{5}4+\frac{1}{3}{\log }_{5}x$ is ${\log }_{5}4\sqrt[3]{x}$ .

Explanation of Solution

Given information:

The expression = ${\log }_{5}4+\frac{1}{3}{\log }_{5}x$

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:

The given expression is $\ln 12-\ln 4$ .

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

  ${\log }_{5}4+\frac{1}{3}{\log }_{5}x={\log }_{5}4+{\log }_5{x}^{\frac{1}{3}}$

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

  $\begin{array}{c}{\log }_{5}4+\frac{1}{3}{\log }_{5}x={\log }_{5}4+{\log }_5{x}^{\frac{1}{3}}\\ ={\log }_{5}4+{\log }_5\sqrt[3]{x}\\ ={\log }_{5}4\sqrt[3]{x}\end{array}$

Hence, the required condensed form of the expression ${\log }_{5}4+\frac{1}{3}{\log }_{5}x$ is ${\log }_{5}4\sqrt[3]{x}$ .