text: # What is the solution of the equation log15 [1/(3^x + x - 5)]=x*(log15 5 - 1) ?
  
  We have to solve log(15) [1/(3^x + x - 5)] = x * (log(15) 5 - 1).
  
  log(15) [1/(3^x + x - 5)] = x * (log(15) 5 - 1)
  
  => log(15) [1/(3^x + x - 5)] = x*log(15) 5 - x
  
  => log(15) [1/(3^x + x - 5)] = x*log(15) 5 - log (15) 15^x
  
  => log(15) [1/(3^x + x - 5)] = log(15) (5^x/ 15^x)
  
  => [1/(3^x + x - 5)] = 5^x/ 15^x
  
  => [1/(3^x + x - 5)] = 1/ 3^x
  
  => 3^x = 3^x + x - 5
  
  => x - 5 = 0
  
  => x = 5
  
  We get x = 5
  
  Approved by eNotes Editorial Team
  
  Posted on
explanation: The document contains a math question and a detailed solution to the equation \( \log_{15} \left[ \frac{1}{3^x + x - 5} \right] = x \left( \log_{15} 5 - 1 \right) \). The solution is provided step-by-step, leading to the final answer \( x = 5 \).
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: # What is the solution of the equation log15 [1/(3^x + x - 5)]=x*(log15 5 - 1) ?
  
  We have to solve log(15) [1/(3^x + x - 5)] = x * (log(15) 5 - 1).
  
  log(15) [1/(3^x + x - 5)] = x * (log(15) 5 - 1)
  
  => log(15) [1/(3^x + x - 5)] = x*log(15) 5 - x
  
  => log(15) [1/(3^x + x - 5)] = x*log(15) 5 - log (15) 15^x
  
  => log(15) [1/(3^x + x - 5)] = log(15) (5^x/ 15^x)
  
  => [1/(3^x + x - 5)] = 5^x/ 15^x
  
  => [1/(3^x + x - 5)] = 1/ 3^x
  
  => 3^x = 3^x + x - 5
  
  => x - 5 = 0
  
  => x = 5
  
  We get x = 5
  
  Approved by eNotes Editorial Team
  
  Posted on
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains a math question and a detailed solution to the equation \( \log_{15} \left[ \frac{1}{3^x + x - 5} \right] = x \left( \log_{15} 5 - 1 \right) \). The solution is provided step-by-step, leading to the final answer \( x = 5 \).
  
  Label: question-and-answer