Chapter 6.6, Problem 16E

To determine

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

Answer to Problem 16E

The required value of $x$ is $0.896$ .

Explanation of Solution

Given Information:

The given equation is $2e^{2x}-7=5$ .

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 $2e^{2x}-7=5$ .

Simplify the above equation.

  $\begin{array}{c}2e^{2x}=5+7\\ 2e^{2x}=12\\ e^{2x}=\frac{12}{2}\\ e^{2x}=6\end{array}$

Take natural log each side with base $e$ of the above equation.

  ${\ln }_e{e}^{2x}={\ln }_{e}6$

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

  $2x{\ln }_{e}e={\ln }_{e}6$

Substitute $1$ for ${\ln }_{e}e$ in the above equations.

  $\begin{array}{c}2x\left(1\right)={\ln }_{e}6\\ 2x={\ln }_{e}6\end{array}$

Substitute $1.792$ for ${\ln }_{e}6$ in the above equation.

  $2x=1.792$

Simplify the above equation.

  $\begin{array}{c}2x=1.792\\ x=\frac{1.792}{2}\\ =0.896\end{array}$

Hence, the required value of $x$ is $0.896$ .