Problem 44

Question

Solve each exponential equation. Express the solution set in terms of natural logarithms or common logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$e^{2 x}-2 e^{x}-3=0$$

Step-by-Step Solution

Verified

Answer

The solution to the exponential equation is $x = \ln3$ or in decimal form, $x \approx 1.10$.

1Step 1 Title: Recognize the Form

The equation $e^{2 x}-2 e^{x}-3=0$ can be seen as an expression similar to a quadratic equation, if you set $u=e^x$. Then, the equation will be in the form $u^2 -2u- 3 = 0$.

2Step 2 Title: Convert Into Quadratic Equation

Substitute $u=e^x$ into the equation, which gives $u^2 -2u -3= 0$.

3Step 3 Title: Solve Quadratic Equation

Solving for $u$, using the quadratic formula $u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, we get $u = 3$ and $u = -1$.

4Step 4 Title: Convert Back to Original Variable

Substitute $e^x$ back in for $u$, giving $e^x = 3$ and $e^x = -1$. Note that there are no solutions for $e^x = -1$ as the exponential function is always positive.

5Step 5 Title: Solve for the Natural Logarithms

Given $e^x = 3$, taking the natural logarithm on both sides of the equation gives $x = \ln3$. Please note that natural log (ln) is the inverse of the exponential function.

6Step 6 Title: Obtain Decimal Approximation

Using a calculator, $\ln3$ is approximately 1.10, correct to two decimal places.

7Step 7 Title: Formulate the Solution Set

The solution set, in terms of natural logarithms, of the exponential equation is $x = \ln3$, and the decimal approximation is $x = 1.10$.

Problem 44

Problem 45