Problem 46 - College Algebra Essentials

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. $$e^{4 x}-3 e^{2 x}-18=0$$

Step-by-Step Solution

Answer

\(x = 0.90\)

Step 1: Rearrange the equation

Notice that the equation \(e^{4 x}-3 e^{2 x}-18=0\) resembles a quadratic equation. Let's substitute \(e^{2x}\) with \(y\). So we have \(y^2 - 3y - 18 = 0\).

Step 2: Factorize quadratic equation

This equation can be factored as \( (y - 6)(y + 3) = 0\). This gives two solutions: \(y_1=6\) and \(y_2=-3\).

Step 3: Sub back the original value and solve for x

Substitute back \(y = e^{2x}\). We have the equations \(e^{2x}=6\) and \(e^{2x}=-3\). It is not possible to have a negative result for \(e^{2x}\). So for \(e^{2x}=6\), we take the natural logarithm on both sides to solve for x. This leads to \(2x = \ln(6)\) and \(x = \frac{1}{2}\ln(6)\).

Step 4: Find the Decimal approximation

To find the decimal approximation, evaluate \(\frac{1}{2}\ln(6)\) using a calculator. This gives the approximate result \(x = 0.90\)