Chapter 3.4, Problem 115E

a.

To determine

To find:

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

Answer to Problem 115E

Our required interval would be $(5,6)$ .

Explanation of Solution

Given:

A logarithmic equation and a table: $\ln (2x)=2.4$ .

x	2	3	4	5	6
$\ln (2x)$

Calculation:

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

  $\begin{array}{l}\ln (2x)\\ \Rightarrow \ln (2\cdot 2)\\ \Rightarrow \ln (4)\\ \Rightarrow 1.38629436\approx 1.386\end{array}$

  $\begin{array}{l}\ln (2x)\\ \Rightarrow \ln (2\cdot 3)\\ \Rightarrow \ln (6)\\ \Rightarrow 1.791759469\approx 1.792\end{array}$

  $\begin{array}{l}\ln (2x)\\ \Rightarrow \ln (2\cdot 4)\\ \Rightarrow \ln (8)\\ \Rightarrow 2.07944154\approx 2.079\end{array}$

  $\begin{array}{l}\ln (2x)\\ \Rightarrow \ln (2\cdot 5)\\ \Rightarrow \ln (10)\\ \Rightarrow 2.302585092\approx 2.303\end{array}$

  $\begin{array}{l}\ln (2x)\\ \Rightarrow \ln (2\cdot 6)\\ \Rightarrow \ln (12)\\ \Rightarrow 2.484906649\approx 2.485\end{array}$

The complete table would be:

x	2	3	4	5	6
$\ln (2x)$	1.386	1.792	2.079	2.303	2.485

Since 2.4 lies between 2.303 and 2.485, therefore our required interval would be $(5,6)$ .

b.

To determine

To graph:

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

Answer to Problem 115E

The solution of our given equation would be $x=5.512$ .

Explanation of Solution

Given:

A logarithmic equation: $\ln (2x)=2.4$ .

Calculation:

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

  

We can see that both lines intersect at $x=5.512$ , therefore, $x=5.512$ is a solution for our given function.

c.

To determine

To solve:

The given equation algebraically.

Answer to Problem 115E

The solution of our given equation would be $x\approx 5.512$ .

Explanation of Solution

Given:

A logarithmic equation: $\ln (2x)=2.4$ .

Calculation:

We will use logarithmic properties to solve our given equation.

  $\begin{array}{l}\ln (2x)=2.4\left(\text{Given equation}\right)\\ 2x=e^{2.4}\left(\text{Converting log to exponential form}\right)\\ 2x=11.0231763806\\ \frac{2x}{2}=\frac{11.0231763806}{2}\left(\text{Dividing both sides by 2}\right)\\ x=5.51158819032\\ x\approx 5.512\left(\text{Rounding to three decimal places}\right)\end{array}$

Therefore, the solution of our given equation would be $x\approx 5.512$ .