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Find the exact solutions, in their simplest form, to the equations (a) 2 ln(2x + 1) - 10 = 0 (b) 3e^x = e^7 - Edexcel - A-Level Maths Pure - Question 3 - 2014 - Paper 5

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Find the exact solutions, in their simplest form, to the equations (a) 2 ln(2x + 1) - 10 = 0 (b) 3e^x = e^7

Worked Solution & Example Answer:Find the exact solutions, in their simplest form, to the equations (a) 2 ln(2x + 1) - 10 = 0 (b) 3e^x = e^7 - Edexcel - A-Level Maths Pure - Question 3 - 2014 - Paper 5

Step 1

(a) 2 ln(2x + 1) - 10 = 0

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Answer

To solve the equation, start by isolating the logarithmic term:

  1. Isolate the Logarithm:

    2 ln(2x + 1) = 10

    Divide both sides by 2:

    ln(2x + 1) = 5

  2. Exponentiate to Remove the Logarithm:

    Apply the exponential function to both sides:

    2x + 1 = e^5

  3. Solve for x:

    2x = e^5 - 1

    x = \frac{e^5 - 1}{2}

Step 2

(b) 3e^x = e^7

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Answer

To find x, follow these steps:

  1. Isolate e^x:

    e^x = \frac{e^7}{3}

  2. Take the Natural Logarithm:

    x = ln\left(\frac{e^7}{3}\right)

    This simplifies to:

    x = 7 - ln(3)

Thus, the exact solutions are:

  • For (a): x=e5−12x = \frac{e^5 - 1}{2}
  • For (b): x=7−ln(3)x = 7 - ln(3)

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