Chapter 4, Problem 87RE

To determine

To Compare: Which is larger.

Answer to Problem 87RE

  ${\log }_{4}258>{\log }_{5}620$

Explanation of Solution

Given:Which is larger

  ${\log }_{4}258\text{ and }{\log }_{5}620$

FORMULA USED:

  $\begin{array}{l}\text{Property of logarithm}\\ {\log }_a{A}^c=c{\log }_{a}A\end{array}$

Given: ${\log }_{4}258\text{ and }{\log }_{5}620$

Consider

  $\begin{array}{l}{\log }_{4}256\\ {\log }_{4}256={\log }_4{4}^4=4law:{\log }_a{A}^c=c{\log }_{a}A\\ thus\\ {\log }_{4}258>\mathrm{4......}(i)\end{array}$

Similarly

  $\begin{array}{l}{\log }_{5}625\\ {\log }_{5}625={\log }_5{5}^4=4law:{\log }_a{A}^c=c{\log }_{a}A\\ thus\\ {\log }_{4}620<\mathrm{4.....}(ii)\end{array}$

From equation $(i)and(ii)$

Combining the two inequalities we get

  ${\log }_{4}258>{\log }_{5}620$

Hence, ${\log }_{4}258>{\log }_{5}620$