Chapter 3.5, Problem 15E

Solution

To find: The solution of the given equation and then check it by substituting it into the original equation.

The exact solution of the given equation $3+2e^{-x}=6$ is $x\approx -0.4055$ .

Given information:

The given equation is $3+2e^{-x}=6$ .

Concept Used:

Step involved in solving the logarithm equation:

Isolate the exponential expression.

Take the natural logarithm on both sides.

Apply power rule of logarithm and then solve.

Calculation:

Subtract $80$ from both sides.

  $\begin{array}{c}3+2e^{-x}-3=6-3\\ 2e^{-x}=3\end{array}$

Divide by $2$ on both sides.

  $\begin{array}{c}{e}^{-x}=\frac{3}{2}\\ e^{-x}=1.5\end{array}$

Take natural logarithms on both sides.

  $\begin{array}{c}\ln e^{-x}=\ln 1.5\\ -x\ln e=\ln 1.5\\ -x=\ln 1.5\left[\ln e=1\right]\\ x=-\ln 1.5\end{array}$

Use the TI-83 calculator for calculating the value of $-\ln 1.5$ .

  $-\ln 1.5\approx -0.4055$

So,

  $x\approx -0.4055$

Now, check the result while substituting the value of $x$ in the given equation.

  $\begin{array}{l}3+2e^{-\left(-0.4055\right)}\stackrel{?}{=}6\\ 3+2e^{0.4055}\stackrel{?}{=}6\end{array}$

Use the TI-83 calculator for calculating the value of $3+2e^{0.4055}$ .

  $\begin{array}{c}3+2e^{0.4055}\approx 6\\ 6=6\left(\text{True}\right)\end{array}$

Hence, the value of $x$ is $\approx -0.4055$ .