Chapter 3.4, Problem 65E

To determine

To find : solution to the equationgraphically and algebraically.

Answer to Problem 65E

The solution to the equation $8e^{-2x/3}=11$ is $\underset{\_}{\boxed{x=-0.478}}$

Explanation of Solution

Given information: $8e^{-2x/3}=11$

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 $f\left(x\right)=8e^{-2x/3}$ and right side of the equation $g\left(x\right)=11$ .

The xcoordinate of point of intersection of two graphs is the solution to the equation $8e^{-2x/3}=11$

Formula Used:

1. $\ln m^n=n\ln m$
2. $\ln e=1$

Calculation:

Divide by 8 on both sides of the equation

  $\frac{8e^{-2x/3}}{8}=\frac{11}{8}$

Simplify fraction on both sides of the equation

  $e^{-2x/3}=\frac{11}{8}$

Take natural logarithm on both sides of the equation

  $\ln e^{-2x/3}=\ln \left(\frac{11}{8}\right)$

Applying the logarithmic property $\ln m^n=n\ln m$ to rewrite the left side of the equation

  $\frac{-2x}{3}\cdot \ln e=\ln \left(\frac{11}{8}\right)$

Replace 1 for $\ln e$

  $\frac{-2x}{3}=\ln \left(\frac{11}{8}\right)$

Multiply $\frac{3}{-2}$ in both sides of the equation

  $\frac{-2x}{3}\cdot \frac{3}{-2}=\ln \left(\frac{11}{8}\right)\cdot \frac{3}{-2}$

Simplify fraction on both sides of the equation

  $x\approx -0.477680597$

Conclusion:

The solution to the equation $8e^{-2x/3}=11$ is $\underset{\_}{\boxed{x=-0.478}}$