Chapter 3.4, Problem 84E

a.

To determine

To find:

An interval containing the solution of the equation by completing the table.

Answer to Problem 84E

Our required interval would be $(7,8)$ .

Explanation of Solution

Given:

A logarithmic equation and a table: $11\left(77-e^{x-4}\right)=264$ .

x	5	6	7	8	9
$11\left(77-e^{x-4}\right)$

Calculation:

To complete the table, we will substitute the values of x in our given expression as shown below:

  $\begin{array}{l}11\left(77-e^{x-4}\right)\\ \Rightarrow 11\left(77-e^{5-4}\right)\\ \Rightarrow 11\left(77-e\right)\\ \Rightarrow 817.098899\approx 817.099\end{array}$

  $\begin{array}{l}11\left(77-e^{x-4}\right)\\ \Rightarrow 11\left(77-e^{6-4}\right)\\ \Rightarrow 11\left(77-e^2\right)\\ \Rightarrow 765.7203829\approx 765.720\end{array}$

  $\begin{array}{l}11\left(77-e^{x-4}\right)\\ \Rightarrow 11\left(77-e^{7-4}\right)\\ \Rightarrow 11\left(77-e^3\right)\\ \Rightarrow 626.0590938\approx 626.059\end{array}$

  $\begin{array}{l}11\left(77-e^{x-4}\right)\\ \Rightarrow 11\left(77-e^{8-4}\right)\\ \Rightarrow 11\left(77-e^4\right)\\ \Rightarrow 246.4203496\approx 246.420\end{array}$

  $\begin{array}{l}11\left(77-e^{x-4}\right)\\ \Rightarrow 11\left(77-e^{9-4}\right)\\ \Rightarrow 11\left(77-e^5\right)\\ \Rightarrow -785.544750\approx -785.545\end{array}$

The complete table would be:

x	5	6	7	8	9
$11\left(77-e^{x-4}\right)$	817.099	765.720	626.059	246.420	$-785.545$

Since 264 lies between 626.059and 246.420, therefore our required interval would be $(7,8)$ .

b.

To determine

To graph:

The given function using a graphing utility and approximate the solution.

Answer to Problem 84E

The solution of our given equation would be $x=7.970$ .

Explanation of Solution

Given:

A logarithmic equation: $11\left(77-e^{x-4}\right)=264$ .

Calculation:

Upon graphing our given function, we will get our required graph as shown below:

  

We can see that both lines intersect at $x=7.970$ , therefore, $x=7.970$ is a solution for our given function.

c.

To determine

To solve:

The given equation algebraically.

Answer to Problem 84E

The solution of our given equation would be $x\approx 7.970$ .

Explanation of Solution

Given:

A logarithmic equation: $11\left(77-e^{x-4}\right)=264$ .

Calculation:

We will use logarithmic properties to solve our given equation.

  $\begin{array}{l}11\left(77-e^{x-4}\right)=264\left(\text{Given equation}\right)\\ 847-11e^{x-4}=264\left(\text{Using distributive property}\right)\\ 847-847-11e^{x-4}=264-847\left(\text{Subtracting }847\text{ from both sides}\right)\\ -11e^{x-4}=-583\\ \frac{-11e^{x-4}}{-11}=\frac{-583}{-11}\left(\text{Dividing both sides by }-11\right)\\ e^{x-4}=53\\ \ln \left(e^{x-4}\right)=\ln (53)\left(\text{Taking natural log on both sides}\right)\\ (x-4)\cdot \ln (e)=\ln (53)\left(\text{Using property ln}\left(x^m\right)=m\ln (x)\right)\\ x-4=3.970291913\left(\text{Using property ln(}e\text{)}=\text{1}\right)\\ x-4+4=3.970291913+4\left(\text{Adding 4 on both sides}\right)\\ x=7.970291913\\ x\approx 7.970\left(\text{Rounding to three decimal places}\right)\end{array}$

Therefore, the solution of our given equation would be $x\approx 7.970$ .