text: # Solve for x the equation e^2x-6*e^x+5=0
  
  justaguide | College Teacher | (Level 2) Distinguished Educator
  
  Posted on
  
  We have to solve e^2x - 6*e^x + 5 = 0 for x.
  
  Let e^x = y
  
  => y^2 - 6y + 5 = 0
  
  => y^2 - 5y - y + 5 = 0
  
  => y(y - 5) - 1(y - 5) = 0
  
  => (y - 1)(y - 5) = 0
  
  => y = 1 and y =5
  
  e^x = 1 and e^x = 5
  
  => x = 0 and x = ln 5
  
  The solution of the equation are x = 0 and x = ln 5
  
  tonys538 | Student, Undergraduate | (Level 1) Valedictorian
  
  Posted on
  
  The equation e^2x-6*e^x+5=0 has to be solved.
  
  e^2x-6*e^x+5=0
  
  = (e^x)^2-6*e^x+5=0
  
  = (e^x)^2-5*e^x - e^x +5=0
  
  = e^x*(e^x - 5) - 1(e^x - 5)=0
  
  = (e^x - 1)*(e^x - 5)=0
  
  e^x - 1 = 0
  
  e^x = 1
  
  This is the case when x = 0
  
  e^x - 5 = 0
  
  e^x = 5
  
  Take the natural logarithm of both the sides
  
  x = ln 5
  
  The solution of the equation e^2x-6*e^x+5=0 is x = 0 and x = ln 5
  
  giorgiana1976 | College Teacher | (Level 3) Valedictorian
  
  Posted on
  
  We'll solve the equation using the technique of substitution.
  
  We'll substitute e^x by the variable t.
  
  e^2x=(e^x)^2=t^2
  
  We'll re-write the equation in t:
  
  t^2 -6t + 5 = 0
  
  t1 = [6+sqrt(36-20)]/2
  
  t1 = (6+4)/2
  
  t1 = 5
  
  t2 = (6-4)/2
  
  t2 = 1
  
  Now, we'll determine x:
  
  e^x = t1
  
  e^x = 5
  
  We'll apply natural logarithms boths sides:
  
  ln e^x = ln 5
  
  x = ln 5
  
  e^x = t2
  
  e^x = 1
  
  But e^0 = 1
  
  e^x = e^0
  
  x = 0
  
  The requested solutions of the equation are: {0 ; ln 5}.
explanation: The document contains a math question and multiple answers to the same question. The question is to solve the equation \( e^{2x} - 6e^x + 5 = 0 \). Three different users provide solutions, all arriving at the same answers: \( x = 0 \) and \( x = \ln 5 \).
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: # Solve for x the equation e^2x-6*e^x+5=0
  
  justaguide | College Teacher | (Level 2) Distinguished Educator
  
  Posted on
  
  We have to solve e^2x - 6*e^x + 5 = 0 for x.
  
  Let e^x = y
  
  => y^2 - 6y + 5 = 0
  
  => y^2 - 5y - y + 5 = 0
  
  => y(y - 5) - 1(y - 5) = 0
  
  => (y - 1)(y - 5) = 0
  
  => y = 1 and y =5
  
  e^x = 1 and e^x = 5
  
  => x = 0 and x = ln 5
  
  The solution of the equation are x = 0 and x = ln 5
  
  tonys538 | Student, Undergraduate | (Level 1) Valedictorian
  
  Posted on
  
  The equation e^2x-6*e^x+5=0 has to be solved.
  
  e^2x-6*e^x+5=0
  
  = (e^x)^2-6*e^x+5=0
  
  = (e^x)^2-5*e^x - e^x +5=0
  
  = e^x*(e^x - 5) - 1(e^x - 5)=0
  
  = (e^x - 1)*(e^x - 5)=0
  
  e^x - 1 = 0
  
  e^x = 1
  
  This is the case when x = 0
  
  e^x - 5 = 0
  
  e^x = 5
  
  Take the natural logarithm of both the sides
  
  x = ln 5
  
  The solution of the equation e^2x-6*e^x+5=0 is x = 0 and x = ln 5
  
  giorgiana1976 | College Teacher | (Level 3) Valedictorian
  
  Posted on
  
  We'll solve the equation using the technique of substitution.
  
  We'll substitute e^x by the variable t.
  
  e^2x=(e^x)^2=t^2
  
  We'll re-write the equation in t:
  
  t^2 -6t + 5 = 0
  
  t1 = [6+sqrt(36-20)]/2
  
  t1 = (6+4)/2
  
  t1 = 5
  
  t2 = (6-4)/2
  
  t2 = 1
  
  Now, we'll determine x:
  
  e^x = t1
  
  e^x = 5
  
  We'll apply natural logarithms boths sides:
  
  ln e^x = ln 5
  
  x = ln 5
  
  e^x = t2
  
  e^x = 1
  
  But e^0 = 1
  
  e^x = e^0
  
  x = 0
  
  The requested solutions of the equation are: {0 ; ln 5}.
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains a math question and multiple answers to the same question. The question is to solve the equation \( e^{2x} - 6e^x + 5 = 0 \). Three different users provide solutions, all arriving at the same answers: \( x = 0 \) and \( x = \ln 5 \).
  
  Label: question-and-answer