Functions: Exponentials and Logarithmic Functions

Exponential growth and decay
Real-life problems such as the spread of diseases, the treatment of pollution, radioactive decay and financial investments can be modelled and solved by exponential and logarithmic functions.

EXAMPLE
The amount A_t micrograms, of a radioactive substance remaining after t years decreases according to the formula A_t = A₀e^−0•003t where A₀ is the amount present initially.
(a) If 250 micrograms are left after 500 years, how many micrograms were present initially?
(b) Find the half-life of the substance.

SOLUTION
(a) We are told that t = 500 and A₅₀₀ = 250,
So A₅₀₀ = A₀e^{−0•003 × 500} = 250
⇒ A₀ = 250e−1∙5

$\frac{250}{e^{-1•5}}$

Using a calculator gives A₀ = 1120 micrograms.
(b) The half-life is the time it takes for the amount of radioactive substance to fall to half of its original amount, so A_t = 12

$\frac{1}{2}$

A₀.
Using A_t = A₀e^−0•003t, we get 12

$\frac{1}{2}$

A₀ = A₀e^−0•003t; so 12

$\frac{1}{2}$

= e^−0•003t.
Taking logs of both sides gives log_e12

$\frac{1}{2}$

= log_ee^−0•003t ⇒ log_e 12

$\frac{1}{2}$

= −0•003tlog_ee
⇒ log_e12

$\frac{1}{2}$

= −0•003t and t = log_e0•5/−0•003 = 231
The half-life is 231 years.

For more on this topic, have a look at our CfE Higher Mathematics Study Guide, pp. 28–29.