Chapter 4.5, Problem 5E

To determine

To calculate: The solution of the exponential , rounded to four decimal places , $e^{-2x}=7$

Answer to Problem 5E

The four decimal places rounded value of exponential expression,

  $x=-0.9730$

Explanation of Solution

Given information:The given expression is as, $e^{-2x}=7$ .

Formula used:

For the exponential expression take log on both sides.

These are the steps to do this ,

  $In{a}^x=xIna,In{e}^x=xIne$

Step 1 take “In” both side.

Step 2. Then to find the value of x one side take all the values of x and another side all the values of “In” and constants .

Step 3. Then with the help of “In” table put the value of “In” .

Step 4. Take the four decimal value of x from there .

Natural logarithms is “In”

Calculation :

Consider the expression, $e^{-2x}=7$

Take logarithms both sides:

  $\begin{array}{l}{e}^{-2x}=7\\ in\left(e^{-2x}\right)=in7\end{array}$

Apply the identity: $In{a}^x=xIna$

  $\begin{array}{c}In{e}^{-2x}=In7\\ -2x=In7\\ x=\frac{1}{2}In7\end{array}$

Apply the value of In7:

Rewrite the expression: $\begin{array}{l}x=-\frac{1}{2}In7\\ x=-0.9730\end{array}$

The above expression can’t be simplified further.

Thus the simplified form of the expression is $x=1.1379$ .