Chapter 6.6, Problem 55E

To determine

To calculate: The inequality ${\log }_{5}x<2$ by algebraically and graphically. Also write which method you prefer and the reason.

Answer to Problem 55E

The required value of $x$ is $0<x<25$ .

Explanation of Solution

Given Information:

The given inequality is ${\log }_{5}x<2$ .

Formula Used:

Property of logarithm:

  $a^{{\log }_{a}x}=x$

Calculation:

Consider the inequality ${\log }_{5}x<2$ .

Solve by algebraically.

  ${\log }_{5}x<2$

Take exponential each side with a base $5$ .

  $5^{{\log }_{5}x}<5^2$

Use property of logarithm in the above equation.

  $\begin{array}{c}x<5^2\\ x<25\end{array}$

Know that, $\log x$ is defined only when $x>0$ .

Thus, the required solution is $0<x<25$ .

Solve by graphically.

Let $y={\log }_{5}x$ and $y=2$ .

Graph for both equations is:

From the intersecting point, $x=25$ .

Thus, the required solution is $0<x<25$ .

It is seen that the algebraic method is more easy to solve than thegraphical method.

Also, it is easy to convert the equation to exponential form and isolates the variable.

So, I prefer algebraically method.

Hence, the required value of $x$ is $0<x<25$ .