Chapter 3.4, Problem 63E

To determine

To find : solution to the equationgraphically and algebraically.

Answer to Problem 63E

The solution to the equation $5^x=212$ is $\underset{\_}{\boxed{x=3.328}}$

Explanation of Solution

Given information: $5^x=212$

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 $f\left(x\right)=5^x$ and right side of the equation $g\left(x\right)=212$ .

The xcoordinate of point of intersection of two graphs is the solution to the equation $5^x=212$

Formula Used:

  $\ln m^n=n\ln m$

Calculation:

Take natural logarithm on both sides of the equation

  $\ln 5^x=\ln 212$

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

  $x\cdot \ln 5=\ln 212$

Divide by $\ln 5$ on both side of the equation

  $\frac{x\cdot \ln 5}{\ln 5}=\frac{\ln 212}{\ln 5}$

Simplify fraction on both side of the equation

  $x\approx 3.32823418$

Conclusion:

The solution to the equation $5^x=212$ is $\underset{\_}{\boxed{x=3.328}}$