Chapter 3.4, Problem 69E

To determine

To find : solution to the equationgraphically and algebraically.

Answer to Problem 69E

The solution to the equation $2\ln \left(x+3\right)=3$ is $\underset{\_}{\boxed{x=1.482}}$

Explanation of Solution

Given information: $2\ln \left(x+3\right)=3$

Concept Involved:

The word “solve” means the process of finding the value of x that makes the equation true. In order to solve we need to undo whatever is done to x . The solution to the equation $f\left(x\right)=g\left(x\right)$ graphically is the x value of the point of intersection of where the two graphs meet.

Graph:

  

Interpretation:

Setting up left side of the equation $f\left(x\right)=2\ln \left(x+3\right)$ and right side of the equation $g\left(x\right)=3$ .

The xcoordinate of point of intersection of two graphs is the solution to the equation $2\ln \left(x+3\right)=3$

Formula Used:

1. If logarithmic equation is $\ln A=m$ , then the equivalent exponential equation is $A=e^m$
2. $\ln e=1$

Calculation:

Divide -4 on both sides of the equation

  $\frac{2\ln \left(x+3\right)}{2}=\frac{3}{2}$

Simplifying fraction on both sides

  $\ln \left(x+3\right)=1.5$

Rewrite the logarithmic equation to exponential equation

  $x+3=e^{1.5}$

Subtracting 3 on both sides of the equation

  $x+3-3=e^{1.5}-3$

Simplifyin right side of the equation

  $x\approx 1.481689$

Conclusion:

The solution to the equation $2\ln \left(x+3\right)=3$ is $\underset{\_}{\boxed{x=1.482}}$