Chapter 3, Problem 40CLT

To determine

To solve the exponential equation:

  $6e^{2x}=72$

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

Answer to Problem 40CLT

1.242

Explanation of Solution

Given:

  $6e^{2x}=72$

Concept Used:

  $u=e^x\iff \ln \left(u\right)=x$

Calculation:

To solve the exponential equation:

  $6e^{2x}=72$

First, simplify the equation as,

  $\begin{array}{c}6e^{2x}=72\\ \Rightarrow e^{2x}=\frac{72}{6}\\ \Rightarrow e^{2x}=12\end{array}$

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

As,

  $\begin{array}{c}{e}^{2x}=12\\ \Rightarrow 2x=\ln \left(12\right)\\ \Rightarrow x=\frac{\ln \left(12\right)}{2}\\ \Rightarrow x\simeq 1.242\end{array}$

Thus, solution of given equation is approximately 1.242.

Now, to verify this result graphically plot the graph of function $6e^{2x}$ and then check whether at x approximately equals to 1.242 value of the function is 72, that is, check whether the point $\left(1.242,72\right)$ lies on the curve of this graph.

  

Since the point $\left(1.242,72\right)$ lies on the graph of the function $6e^{2x}$ , thus the result is verified.