Chapter 6.6, Problem 50E

To determine

To calculate: the value of $x$ .

Answer to Problem 50E

The required value of $x$ is $0<x<256$ .

Explanation of Solution

Given Information:

The given inequality is ${\log }_{4}x<4$ .

Formula Used:

Property of logarithm:

  $a^{{\log }_{a}x}=x$

Calculation:

Consider theinequality is ${\log }_{4}x<4$ .

Convert the above inequality into exponential form.

  ${\log }_{4}x<4$

Take exponential in each side with base $4$ .

  $4^{{\log }_{4}x}<4^4$

Now, use property of logarithm $a^{{\log }_{a}x}=x$ of the aboveinequality.

  $\begin{array}{c}x{\log }_{4}4<4^4\\ x\left(1\right)<256\\ x<256\end{array}$

Since, it is known that $\log x$ is only defined for $x>0$ .

Thus, the solution of the given inequality is $0<x<256$ .

Hence, the required value of $x$ is $0<x<256$ .