Module 4 – Exponential and Logarithmic Functions - Introductory Examples

Radioactive Decay

 

Radioactive materials typically decay at a rate proportional to the amount present, so the time taken for the level of radioactivity to be reduced to half of what it was is independent of the starting level. This time is called the half-life of the material. Suppose that a particular material has a half-life of one week and a sample of this material has a radiation level now of 10 units, then over one week the radiation level will fall to 5 units.  In each subsequent week the radiation level gets halved again, so, after t weeks it will have been halved t times, i.e. multiplied by   .

Thus the radiation level after t weeks will be given by , and if the initial level was not 10 but some other level, say , then that number would replace the 10 in the above formula.

 

Here, the formula for R(t) has the variable t as the exponent (‘upstairs’ part) in a power expression.  Up to now, we have only considered power expressions in which the exponents are constants (e.g. polynomials like  ), but in this unit we study functions defined by equations involving variables in the exponents. Such functions are called exponential functions (or logarithmic functions if it is the dependent variable in the exponent).