Chapter 6.6, Problem 14E

To determine

To calculate: .The equation for the value of $x$ .

Answer to Problem 14E

The required value of $x$ is $0.253$ .

Explanation of Solution

Given Information:

The given equation is ${11}^{6x}=38$ .

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 the equation is ${11}^{6x}=38$ .

Take common logarithm each side with base $10$ .

  ${\log }_{10}{11}^{6x}={\log }_{10}38$

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

  $6x{\log }_{10}11={\log }_{10}38$

Divide each side by ${\log }_{10}11$ in the above equation.

  $\begin{array}{c}\frac{6x{\log }_{10}11}{{\log }_{10}11}=\frac{{\log }_{10}38}{{\log }_{10}11}\\ 6x=\frac{{\log }_{10}38}{{\log }_{10}11}\end{array}$

Substitute $1.579$ for ${\log }_{10}38$ and $1.041$ for ${\log }_{10}11$ in the above equations.

  $6x=\frac{1.579}{1.041}$

Simplify the above equation.

  $\begin{array}{c}6x=\frac{1.579}{1.041}\\ 6x=1.516\\ x=\frac{1.516}{6}\\ x=0.253\end{array}$

Hence, the required value of $x$ is $0.253$ .