A-level · Exponentials & logarithms

Exponential functions and eExam-style questions

Functions with the variable in the power, y = a^x — their graph shape, the asymptote and intercept — and the special exponential e^x whose gradient equals itself.

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

DifficultyAllLevel 3Level 4
  1. 1
    (a) Solve e2x=5e^{2x} = 5, giving your answer in exact form.
    (b) Explain why the equation e2x=−1e^{2x} = -1 has no real solutions.
    Level 3 · 3 marksTry it
  2. 2
    Solve the equation

    e2x−7ex+10=0\mathrm{e}^{2x} - 7\mathrm{e}^{x} + 10 = 0

    giving your answers exactly. (4 marks)
    Level 3 · 4 marksTry it
  3. 3
    f(x)=e2x−8ex+12f(x) = e^{2x} - 8e^x + 12.
    (a) Solve the equation f(x)=0f(x) = 0, giving your answers exactly. (4 marks)
    (b) Find the exact coordinates of the stationary point of the curve y=f(x)y = f(x), and state its nature. (4 marks)
    (c) Show that ∫0ln⁡2f(x)  dx=12ln⁡2−132\displaystyle\int_0^{\ln 2} f(x)\; dx = 12\ln 2 - \frac{13}{2}. (3 marks)
    Level 4 · 11 marksTry it