Chapter 3.4, Problem 67E

To determine

To find : solution to the equationgraphically and algebraically.

Answer to Problem 67E

The solution to the equation $3-\ln x=0$ is $\underset{\_}{\boxed{x=20.086}}$

Explanation of Solution

Given information: $3-\ln x=0$

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 $f\left(x\right)=3-\ln x$ and right side of the equation $g\left(x\right)=0$ .

The xcoordinate of point of intersection of two graphs is the solution to the equation $3-\ln x=0$

Formula Used:

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

Calculation:

Subtract 3 on both sides of the equation

  $3-\ln x-3=0-3$

Simplify fraction on both sides

  $-\ln x=-3$

Divide -1 on both sides of the equation

  $\frac{-\ln x}{-1}=\frac{-3}{-1}$

Simplifying fraction on both sides

  $\ln x=3$

Rewrite the logarithmic equation to exponential equation

  $x=e^3$

Simplifyin right side of the equation

  $x\approx 20.08553692$

Conclusion:

The solution to the equation $3-\ln x=0$ is $\underset{\_}{\boxed{x=20.086}}$