Chapter 6.6, Problem 48E

To determine

To calculate: the value of $x$ .

Answer to Problem 48E

The required value of $x$ is $x\le 2.58$ .

Explanation of Solution

Given Information:

The given inequality is $4^x\le 36$ .

Formula Used:

Power rule of logarithmic functions:

  ${\log }_a{x}^p=p{\log }_{a}x$

Where $a$ is the base, $x$ is the variable and $p$ is the power of variable.

Calculation:

Consider theinequality is $4^x\le 36$ .

Take logarithm with base $4$ each side of the aboveinequality.

  ${\log }_4{4}^x\le {\log }_{4}36$

Now, use power rule of logarithm ${\log }_a{x}^p=p{\log }_{a}x$ in the aboveinequality.

  $x{\log }_{4}4\le {\log }_{4}36$

Now, use property of logarithm ${\log }_{a}a=1$ of the aboveinequality.

  $\begin{array}{c}x{\log }_{4}4\le {\log }_{4}36\\ x\left(1\right)\le {\log }_{4}36\\ x\le {\log }_{4}36\end{array}$

Simplify the above equation $x\le {\log }_{4}36$ .

  $x\le {\log }_{4}36$

Substitute $2.58$ for ${\log }_{4}36$ in theinequality.

  $x\le 2.58$

Hence, the required value of $x$ is $x\le 2.58$ .