Chapter 3.4, Problem 16E

To determine

To find : solution to the equation $f\left(x\right)=g\left(x\right)$ algebraically and graphically.

Answer to Problem 16E

The solution to the equation $f\left(x\right)=g\left(x\right)$ is $\underset{\_}{\boxed{x=9}}$

Explanation of Solution

Given information: $f\left(x\right)={\log }_{3}x,g\left(x\right)=2$

  

Concept Involved:

The solution to the equation $f\left(x\right)=g\left(x\right)$ graphically is thex- value of the point of intersection of where the two graphs meet.

Calculation:

The solution is the point of intersection of graph of $f\left(x\right)\&g\left(x\right)$ , so $\left(8,3\right)$ is the solution

  

Set up an equation $f\left(x\right)=g\left(x\right)$ substituting $f\left(x\right)={\log }_{3}x,g\left(x\right)=2$

  ${\log }_{3}x=2$

Rewriting the logarithmic equation as an exponential equation using the rule which states “if logarithmic equation is ${\log }_{b}A=m$ , then equivalent exponential equation is given by $A=b^m$ ”

  $x=3^2$

Simplifying the right side of the equation we get

  $x=9$

Conclusion:

The solution to the equation $f\left(x\right)=g\left(x\right)$ is $\underset{\_}{\boxed{x=9}}$