A-level · Exponentials & logarithms

Solving exponential and logarithmic equationsExam-style questions

Taking logs to solve a^x = b, equations with e^x and ln, hidden quadratics in e^x or a^x, and log equations solved with the laws — always checking which roots are valid.

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14 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAllLevel 2Level 3Level 4
  1. 1
    Solve the equation

    8x+1=42x−18^{x+1} = 4^{2x-1}

    (4 marks)
    Level 2 · 4 marksTry it
  2. 2
    Find the set of values of xx for which 7x<32x−17^{x} < 3^{2x - 1}. Give your answer in a simplified exact form.

    (3 marks)
    Level 2 · 3 marksTry it
  3. 3
    Using logarithms, solve the equation

    72x−1=3x+4,7^{2x - 1} = 3^{x + 4},

    giving your answer correct to 3 significant figures.

    (3 marks)
    Level 2 · 3 marksTry it
  4. 4
    Solve the equation

    32x+1=5x+23^{2x+1} = 5^{x+2},

    giving your answer in the form x=ln⁡aln⁡bx = \dfrac{\ln a}{\ln b}, where aa and bb are rationals, and then to 3 significant figures. (5 marks)
    Level 3 · 5 marksTry it
  5. 5
    Solve the equation ln⁡(2x+3)=2ln⁡x\ln(2x+3) = 2\ln x. (4 marks)
    Level 3 · 4 marksTry it

9 more questions on solving exponential and logarithmic equations

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