Solve the following exponential equation. Express the solution set in terms of natural logarithms or common logarithms. Then, use a $$ e^{2 x}-3 e^{x}+2=0 $$ The solution set expressed in terms of logarithms is $$ \{0, \ln 2\} $$. Use a comma to separate answers as needed. Simplify your answer. Use integers or fractions for any numbers in the expression. Use Now use a calculator to obtain a decimal approximation for the solution. The solution set is $$ \left\{\begin{array}{l}\text {. }\end{array}\right. $$ (Use a comma to separate answers as needed. Round to two decimal places as needed.)

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UpStudy AI · Accepted Answer

We start with the given equation:
$$
e^{2x} - 3e^x + 2 = 0.
$$

**Step 1.** Let $$ u = e^x $$. Since $$ e^x > 0 $$, we have:
$$
u^2 - 3u + 2 = 0.
$$

**Step 2.** Factor the quadratic:
$$
(u - 1)(u - 2) = 0.
$$
Thus, the solutions for $$ u $$ are:
$$
u = 1 \quad \text{or} \quad u = 2.
$$

**Step 3.** Convert back to $$ x $$ by recalling that $$ u = e^x $$.

For $$ u = 1 $$:
$$
e^x = 1 \quad \Longrightarrow \quad x = \ln 1 = 0.
$$

For $$ u = 2 $$:
$$
e^x = 2 \quad \Longrightarrow \quad x = \ln 2.
$$

So the solution set in terms of logarithms is:
$$
\{0, \ln 2\}.
$$

**Step 4.** Now, using a calculator, we obtain a decimal approximation for $$ \ln 2 $$:
$$
\ln 2 \approx 0.69.
$$

Thus, the solution set with decimal approximations is:
$$
\{0.00, 0.69\}.
$$
The solution set is $$ \{0, 0.69\} $$.

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