Chapter 8.4, Problem 59PPSE

Solution

To evaluate: The logarithmic expression ${\log }_5\left(5\sqrt{5}\right)$ .

The evaluated value of ${\log }_5\left(5\sqrt{5}\right)=\frac{3}{2}$ .

Given:

The logarithmic expression to evaluate is ${\log }_5\left(5\sqrt{5}\right)$ .

Concept used:

Use the logarithm result ${\log }_a{\left(m\right)}^n=n{\log }_a\left(m\right)\text{ and }{\log }_a\left(a\right)=1$ to evaluate the logarithmic expression.

Calculation:

With the help of the concept stated above, the logarithmic expression ${\log }_5\left(5\sqrt{5}\right)$ is evaluated as:

  $\begin{array}{c}{\log }_5\left(5\sqrt{5}\right)={\log }_5\left(5\cdot \sqrt{5}\right)\\ ={\log }_5\left(5\cdot 5^{\frac{1}{2}}\right)\\ ={\log }_5{\left(5\right)}^{\frac{3}{2}}\\ =\left(\frac{3}{2}\right){\log }_5\left(5\right)\mathrm{....}\left(\because {\log }_a{\left(m\right)}^n=n{\log }_a\left(m\right)\right)\\ =\frac{3}{2}\left(1\right)\mathrm{....}\left(\because {\log }_a\left(a\right)=1\right)\\ =\frac{3}{2}\end{array}$

Conclusion:

The evaluated value of ${\log }_5\left(5\sqrt{5}\right)=\frac{3}{2}$ .