Chapter 3, Problem 43CLT

To determine

To solve the exponential equation:

  $2500e^{0.05x}=500,000$

Round the result to three decimal places and to verify the result graphically.

Answer to Problem 43CLT

105.966

Explanation of Solution

Given:

  $2500e^{0.05x}=500,000$

Concept Used:

  $u=e^x\iff \ln \left(u\right)=x$

Calculation:

To solve the exponential equation:

  $2500e^{0.05x}=500,000$

First, simplify the equation as,

  $\begin{array}{c}2500e^{0.05x}=500,000\\ \Rightarrow e^{0.05x}=\frac{500,000}{2500}\\ \Rightarrow e^{0.05x}=200\end{array}$

Now rewrite it in logarithmic equation using the property $u=e^x\iff \ln \left(u\right)=x$

As,

  $\begin{array}{c}{e}^{0.05x}=200\\ \Rightarrow 0.05x=\ln \left(200\right)\\ \Rightarrow x=\frac{\ln \left(200\right)}{0.05}\\ \Rightarrow x\simeq \frac{5.298}{0.05}\\ \Rightarrow x\simeq 105.966\end{array}$

Thus, solution of given equation is approximately 105.966.

Now, to verify this result graphically plot the graph of function $2500e^{0.05x}$ and then check whether at x approximately equals to 105.966 value of the function is 500000, that is, check whether the point $\left(105.966,500000\right)$ lies on the curve of this graph.

  

Since the point $\left(105.966,500000\right)$ lies on the graph of the function $2500e^{0.05x}$ , thus the result is verified.