Chapter 3.4, Problem 70E

To determine

To solve:

The given equation algebraically. Round final answer to three decimal places. Use a graphing utility to verify your answer.

Answer to Problem 70E

The solution of our given equation is $x\approx 180.070$ .

Explanation of Solution

Given:

An exponential equation: $250\left[\frac{{\left(1+0.01\right)}^x}{0.01}\right]=150,000$ .

Calculation:

We will use logarithmic properties to solve our given equation as shown below:

  $\begin{array}{l}250\left[\frac{{\left(1+0.01\right)}^x}{0.01}\right]=150,000\left(\text{Given equation}\right)\\ \frac{0.01}{250}\cdot 250\left[\frac{{\left(1.01\right)}^x}{0.01}\right]=\frac{0.01}{250}\cdot 150,000\left(\text{Making a common denominator}\right)\\ {\left(1.01\right)}^x=6\\ \ln \left({\left(1.01\right)}^x\right)=\ln \left(6\right)\left(\text{Taking natural log on both sides}\right)\\ x\cdot \ln \left(1.01\right)=\ln \left(6\right)\left(\text{Using log property ln}\left(a^m\right)=m\ln (a)\right)\\ \frac{x\cdot \ln \left(1.01\right)}{\ln \left(1.01\right)}=\frac{\ln \left(6\right)}{\ln \left(1.01\right)}\left(\text{Dividing both sides by }\ln \left(1.01\right)\right)\\ x=180.070340943\\ x\approx 180.070\left(\text{Rounding to three decimal places}\right)\end{array}$

Upon graphing our given function, we will get our required graph as shown below:

  

Upon looking at our given graph, we can see that the solution of our given equation is $x\approx 180.070$ .