Find real root of equation x+ log base 10 x = 3.375 up to 5 decimal using newton raphson method

Mathematics
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find real root of equation x+ log base 10 x = 3.375 up to 5 decimal using newton raphson method

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Find real root of equation x + \log_{10} x = 3.375 up to 5 decimal using Newton-Raphson method

Answer:

To find the real root of the equation x + \log_{10} x = 3.375 up to 5 decimal places using the Newton-Raphson method, follow these steps:

  1. Define the function and its derivative:
    Let f(x) = x + \log_{10} x - 3.375.
    The derivative of f(x), f'(x), is needed for the Newton-Raphson iteration.

    We have:

    f(x) = x + \log_{10} x - 3.375

    The derivative:

    f'(x) = 1 + \frac{1}{x \ln(10)}

  2. Initialize and set the iteration formula:
    Choose an initial guess ( x_0 ). For the Newton-Raphson method, the iteration formula is:

    x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}

    We can start with an initial guess. Let’s try ( x_0 = 3 ).

  3. Execute the Newton-Raphson iterations:

    • First iteration:

      f(3) = 3 + \log_{10} 3 - 3.375 \approx -0.225309
      f'(3) = 1 + \frac{1}{3 \ln(10)} \approx 1.144716
      x_1 = 3 - \frac{-0.225309}{1.144716} \approx 3.196801

    • Second iteration:

      f(3.196801) = 3.196801 + \log_{10} 3.196801 - 3.375 \approx 0.025349
      f'(3.196801) = 1 + \frac{1}{3.196801 \ln(10)} \approx 1.135920
      x_2 = 3.196801 - \frac{0.025349}{1.135920} \approx 3.174496

    • Third iteration:

      f(3.174496) = 3.174496 + \log_{10} 3.174496 - 3.375 \approx 0.000123
      f'(3.174496) = 1 + \frac{1}{3.174496 \ln(10)} \approx 1.136978
      x_3 = 3.174496 - \frac{0.000123}{1.136978} \approx 3.174388

    • Fourth iteration (for higher precision):

      f(3.174388) = 3.174388 + \log_{10} 3.174388 - 3.375 \approx 9.7 \times 10^{-8}
      f'(3.174388) = 1 + \frac{1}{3.174388 \ln(10)} \approx 1.136982
      x_4 = 3.174388 - \frac{9.7 \times 10^{-8}}{1.136982} \approx 3.174388

  4. Verify the final result:
    We reach ( x = 3.174388 ) within the precision limit.

Final Answer:
The real root of the equation x + \log_{10} x = 3.375 up to 5 decimal places using the Newton-Raphson method is approximately ( \boxed{3.17439} ).