Chapter 3.4, Problem 70E

To determine

To find : solution to the equationgraphically and algebraically.

Answer to Problem 70E

The solution to the equation $\ln \left(x+1\right)=2-\ln x$ is $\underset{\_}{\boxed{x\approx 2.264}}$

Explanation of Solution

Given information: $\ln \left(x+1\right)=2-\ln x$

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 $f\left(x\right)=\ln \left(x+1\right)$ and right side of the equation $g\left(x\right)=2-\ln x$ .

The xcoordinate of point of intersection of two graphs is the solution to the equation $\ln \left(x+1\right)=2-\ln x$

Formula Used:

1. If logarithmic equation is $\ln A=m$ , then the equivalent exponential equation is $A=e^m$
2. $\ln e=1$
3. $\ln m+\ln n=\ln \left(m\cdot n\right)$

Calculation:

In the process of getting the x terms in one side of the equation we need to add $lnx$ on both sides$\ln \left(x+1\right)+\ln x=2-\ln x+\ln x$

Simplifying on both sides of the equation

  $\ln \left(x+1\right)+\ln x=2$

Use the logarithmic property $\ln m+\ln n=\ln \left(m\cdot n\right)$ to rewrite the left side of the equation

  $\ln \left(x\cdot \left(x+1\right)\right)=2$

Rewrite the logarithmic equation to exponential equation

  $x\left(x+1\right)=e^2$

Distribute in left side of the equation

  $x^2+x=e^2$

Subtracting $e^2$ on both sides of the equation

  $x^2+x-e^2=e^2-e^2$

Simplifying on both sides of the equation

  $x^2+x-e^2=0$

When the quadratic equation of the form $a{x}^2+bx+c=0$ , we can use the quadratic formula $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

  $x=\frac{-1\pm \sqrt{1^2-4\left(1\right)\left(-e^2\right)}}{2}$

Find the two values of x by simplifying further

$x=\frac{-1+\sqrt{1+4e^2}}{2}$$x\approx 2.26388424$

$\frac{-1-\sqrt{1+4e^2}}{2}$$x\approx -3.26388424$

Check for extraneous solution by substituting the result in the original equation.

  $x\approx -3.26388424$ makes the original equation FALSE.

  $x\approx 2.26388424$ makes the original equation TRUE.

Conclusion:

The solution to the equation $\ln \left(x+1\right)=2-\ln x$ is $\underset{\_}{\boxed{x\approx 2.264}}$