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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. Add 10 to both sides: 2ln⁡(2x+1)=102 \ln(2x + 1) = 10

  2. Divide both sides by 2: ln⁡(2x+1)=5\ln(2x + 1) = 5

  3. Exponentiate both sides to eliminate the logarithm: 2x+1=e52x + 1 = e^5

  4. Subtract 1 from both sides: 2x=e5−12x = e^5 - 1

  5. Finally, divide by 2: x=e5−12x = \frac{e^5 - 1}{2}

This gives us the exact solution for part (a).

Step 2

(b) 3e^x = e^7

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Answer

To solve this equation, follow these steps:

  1. Divide both sides by 3: ex=e73e^x = \frac{e^7}{3}

  2. Take the natural logarithm of both sides: x=ln⁡(e73)x = \ln\left(\frac{e^7}{3}\right)

  3. By applying the properties of logarithms: x=ln⁡(e7)−ln⁡(3)x = \ln(e^7) - \ln(3) x=7−ln⁡(3)x = 7 - \ln(3)

Thus, the exact solution for part (b) is: x=7−ln⁡(3)x = 7 - \ln(3)

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