Exponential and logarithmic functions

  • Harmonically
    problem
    Favourite

    Harmonically

    Age
    16 to 18
    Challenge level

    Is it true that a large integer m can be taken such that: 1 + 1/2 + 1/3 + ... +1/m > 100 ?

  • Blood buffers
    problem
    Favourite

    Blood Buffers

    Age
    16 to 18
    Challenge level

    Investigate the mathematics behind blood buffers and derive the form of a titration curve.

  • Tuning and Ratio
    problem

    Tuning and Ratio

    Age
    16 to 18
    Challenge level
    Why is the modern piano tuned using an equal tempered scale and what has this got to do with logarithms?
  • Prime counter
    problem

    Prime Counter

    Age
    16 to 18
    Challenge level
    A short challenge concerning prime numbers.
  • Sierpinski Triangle
    problem

    Sierpinski Triangle

    Age
    16 to 18
    Challenge level
    What is the total area of the triangles remaining in the nth stage of constructing a Sierpinski Triangle? Work out the dimension of this fractal.
  • Stirling work
    problem

    Stirling Work

    Age
    16 to 18
    Challenge level
    See how enormously large quantities can cancel out to give a good approximation to the factorial function.
  • Complex Sine
    problem

    Complex Sine

    Age
    16 to 18
    Challenge level
    Solve the equation sin z = 2 for complex z. You only need the formula you are given for sin z in terms of the exponential function, and to solve a quadratic equation and use the logarithmic function.
  • Big, Bigger, Biggest
    problem

    Big, Bigger, Biggest

    Age
    16 to 18
    Challenge level

    Which is the biggest and which the smallest of $2000^{2002}, 2001^{2001} \text{and } 2002^{2000}$?

  • \pH temperature
    problem

    pH Temperature

    Age
    16 to 18
    Challenge level

    At what temperature is the pH of water exactly 7?

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