text: # Solve $\log_{5-x} {(x^2 -2x+65)}$ $=$ $2$ and Find the value of $x$
  
  $x^2 -2x+65$ and $5-x$ are two algebraic expressions. The value of logarithm of $x^2 -2x+65$ to $5-x$ is $2$. The value of $x$ can be determined by solving this logarithmic equation.
  
  $\log_{5-x} {(x^2 -2x+65)} = 2$
  
  The logarithmic equation can be expressed in exponential form by the relation between logarithms and exponential functions.
  
  $\implies x^2 -2x+65 = {(5-x)}^2$
  
  Expand the right hand side expression by the square of difference of two terms.
  
  $\implies x^2 -2x+65 = 5^2 + x^2 -2 \times 5 \times x$
  
  $\implies x^2 -2x+65 = x^2 -10x +25$
  
  $\implies x^2 -x^2 -2x +10x = -65+25$
  
  $\implies \require{cancel} \cancel{x^2} -\cancel{x^2} +8x = -40$
  
  $\implies 8x = -40$
  
  $\implies x = \dfrac{-40}{8}$
  
  $\implies \require{cancel} x = -\dfrac{\cancel{40}}{\cancel{8}}$
  
  $\therefore \,\,\,\,\,\, x = -5$
  
  Therefore, it is solved that the value of $x$ is $-5$ and it is the required solution for this logarithmic problem in algebraic form.
  
  ## Verification
  
  The value of $x$ can be verified mathematically by substituting the variable $x$ by $-5$ in the left hand side logarithmic expression. If the logarithmic expression is given the right hand side value and it is $2$, then the value of $x = -5$ is true.
  
  $\log_{5-x} {(x^2 -2x+65)}$
  
  Put $x = -5$ and simplify the expression.
  
  $= \log_{(5-(-5)} {[{(-5)}^2 -2(-5)+65]}$
  
  $= \log_{10} {(25+10+65)}$
  
  $= \log 100$
  
  $= \log 10^2$
  
  $= 2$
  
  Therefore, the value of $x$ is $-5$ and it is required solution for this maths problem.
explanation: The document contains a math question and its solution. The question is to solve the logarithmic equation \(\log_{5-x} {(x^2 -2x+65)} = 2\) and find the value of \(x\). The solution is provided, and it is verified that \(x = -5\) is the correct solution.
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 $\log_{5-x} {(x^2 -2x+65)}$ $=$ $2$ and Find the value of $x$
  
  $x^2 -2x+65$ and $5-x$ are two algebraic expressions. The value of logarithm of $x^2 -2x+65$ to $5-x$ is $2$. The value of $x$ can be determined by solving this logarithmic equation.
  
  $\log_{5-x} {(x^2 -2x+65)} = 2$
  
  The logarithmic equation can be expressed in exponential form by the relation between logarithms and exponential functions.
  
  $\implies x^2 -2x+65 = {(5-x)}^2$
  
  Expand the right hand side expression by the square of difference of two terms.
  
  $\implies x^2 -2x+65 = 5^2 + x^2 -2 \times 5 \times x$
  
  $\implies x^2 -2x+65 = x^2 -10x +25$
  
  $\implies x^2 -x^2 -2x +10x = -65+25$
  
  $\implies \require{cancel} \cancel{x^2} -\cancel{x^2} +8x = -40$
  
  $\implies 8x = -40$
  
  $\implies x = \dfrac{-40}{8}$
  
  $\implies \require{cancel} x = -\dfrac{\cancel{40}}{\cancel{8}}$
  
  $\therefore \,\,\,\,\,\, x = -5$
  
  Therefore, it is solved that the value of $x$ is $-5$ and it is the required solution for this logarithmic problem in algebraic form.
  
  ## Verification
  
  The value of $x$ can be verified mathematically by substituting the variable $x$ by $-5$ in the left hand side logarithmic expression. If the logarithmic expression is given the right hand side value and it is $2$, then the value of $x = -5$ is true.
  
  $\log_{5-x} {(x^2 -2x+65)}$
  
  Put $x = -5$ and simplify the expression.
  
  $= \log_{(5-(-5)} {[{(-5)}^2 -2(-5)+65]}$
  
  $= \log_{10} {(25+10+65)}$
  
  $= \log 100$
  
  $= \log 10^2$
  
  $= 2$
  
  Therefore, the value of $x$ is $-5$ and it is required solution for this maths problem.
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains a math question and its solution. The question is to solve the logarithmic equation \(\log_{5-x} {(x^2 -2x+65)} = 2\) and find the value of \(x\). The solution is provided, and it is verified that \(x = -5\) is the correct solution.
  
  Label: question-and-answer