Chapter 6.6, Problem 47E

To determine

To calculate: the value of $x$ .

Answer to Problem 47E

The required value of $x$ is $x>1.8$ .

Explanation of Solution

Given Information:

The given inequality is $9^x>54$ .

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.

Property of logarithm:

  ${\log }_{a}a=1$

Calculation:

Consider the inequality is $9^x>54$ .

Take logarithm each side with base $9$ of the above inequality.

  ${\log }_9{9}^x>{\log }_{9}54$

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

  $x{\log }_{9}9>{\log }_{9}54$

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

  $\begin{array}{c}x{\log }_{9}9>{\log }_{9}54\\ x\left(1\right)>{\log }_{9}54\\ x>{\log }_{9}54\end{array}$

Simplify the above inequality.

  $x>{\log }_{9}54$

Substitute $1.8$ for ${\log }_{9}54$ in the above inequality.

  $x>1.8$

Hence, the required value of $x$ is $x>1.8$ .