Solve the following exponential equation. Express the solution set in terms of natural logarithms or common logarithms. Then, us $$ e^{2 x}-4 e^{x}+3=0 $$ The solution set expressed in terms of logarithms is $$ \{\square $$. Use a comma to separate answers as needed. Simplify $$ \} $$.

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UpStudy AI · Accepted Answer

We start with the equation  
$$
e^{2x} - 4e^{x} + 3 = 0.
$$

**Step 1. Substitute:**  
Let  
$$
u = e^x.
$$  
Then, since $$ e^{2x} = \left(e^x\right)^2 $$, the equation becomes  
$$
u^2 - 4u + 3 = 0.
$$

**Step 2. Factor the quadratic:**  
Factorizing, we have  
$$
u^2 - 4u + 3 = (u-1)(u-3) = 0.
$$  
Thus,  
$$
u = 1 \quad \text{or} \quad u = 3.
$$

**Step 3. Substitute back for $$ x $$:**  
Recall that $$ u = e^x $$. So:

- For $$ u = 1 $$:  
  $$
  e^x = 1 \quad \Rightarrow \quad x = \ln(1) = 0.
  $$

- For $$ u = 3 $$:  
  $$
  e^x = 3 \quad \Rightarrow \quad x = \ln(3).
  $$

**Final Answer:**  
The solution set expressed in terms of natural logarithms is  
$$
\{0,\,\ln(3)\}.
$$
The solution set is $$ \{0, \ln(3)\} $$.

Solve the following exponential equation. Express the solution set in terms of natural logarithms or common logarithms. Then, us \[ e^{2 x}-4 e^{x}+3=0 \] The solution set expressed in terms of logarithms is \( \{\square \). Use a comma to separate answers as needed. Simplify \( \} \).

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