Problem 43 - 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. $$e^{2 x}-3 e^{x}+2=0$$

Step-by-Step Solution

Answer

The solutions to the equation are \(x = ln(1)\) and \(x = ln(2)\), which approximate to \(x \approx 0.00\) and \(x \approx 0.69\), respectively when rounded off to two decimal places.

Step 1: Substitution for Quadratic Form

Substitute \(e^x\) with \(u\), such that \(e^{2x}\) becomes \(u^{2}\), \(e^x\) becomes \(u\), thus the equation can be written in the form \(u^2 - 3u + 2 = 0\).

Step 2: Solve the Quadratic Equation

Factor the quadratic equation, \(u^2 - 3u + 2 = 0\), which results in \((u-1)(u-2) = 0\). Setting these factors equal to zero gives the roots as \(u_1 = 1\) and \(u_2 = 2\).

Step 3: Transform the Solution to Original Variable x

Since \(u = e^x\), our solutions transform to \(e^x = 1\) and \(e^x = 2\). Applying the natural logarithm to each side of these equations gives \(x = ln(1)\) and \(x = ln(2)\).

Step 4: Decimal Approximation of the Solution

Finally, use a calculator to find the decimal approximation to the solutions. We get \(x \approx 0.00\) for \(x = ln(1)\) and \(x \approx 0.69\) for \(x = ln(2)\), each correct to the two decimal places.