/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 43 Solve each exponential equation.... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 43

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

Short Answer

Expert verified
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 by step solution

01

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 \(u^2 - 3u + 2 = 0\).
02

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 \(u_2 = 2\).
03

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 \(x = ln(2)\).
04

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.

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