Exponential and logarithmic functions

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.
problem
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 ?
problem
The legacy

Age

14 to 16

Challenge level

Your school has been left a million pounds in the will of an ex- pupil. What model of investment and spending would you use in order to ensure the best return on the money?
problem
Dating made easier

Age

14 to 16

Challenge level

If a sum invested gains 10% each year how long before it has doubled its value?
problem
Back fitter

Age

14 to 18

Challenge level

10 graphs of experimental data are given. Can you use a spreadsheet to find algebraic graphs which match them closely, and thus discover the formulae most likely to govern the underlying processes?
problem
What's that graph?

Age

14 to 18

Challenge level

Can you work out which processes are represented by the graphs?
problem
Big, bigger, biggest

Age

16 to 18

Challenge level

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

Age

16 to 18

Challenge level

The equation a^x + b^x = 1 can be solved algebraically in special cases but in general it can only be solved by numerical methods.
problem
Power match

Age

16 to 18

Challenge level

Can you locate these values on this interactive logarithmic scale?
$Mixing \pH$

problem
Mixing pH

Age

16 to 18

Challenge level

Use the logarithm to work out these pH values