Math Help

In a Nut Shell: Different models exists for the prediction of growth or decay. One

model for natural phenomena expresses that the growth or decay is proportional to the

rate of the size of the phenomena. This model can be represented by the following

differential equation:

dy/dt = k y (1)

where y(t) is population or size at any time t

k is the growth or decay rate i.e. k may be positive or negative.

If k > 0 the function, y(t), increases with time and k represents growth rate

and if k < 0 the function, y(t), decreases with time, t. and k represents the decay rate

Integration of eq (1) is by separation of variables and results in the following exponential

equation for y(t).

y(t) = C e ^kt (2)

where y(t) is the size of the population at any time t

C is the initial size (at t = 0) of the population

k is either the growth or decay rate

Growth Applications may include population increase such as bacteria and cell growth,

or financial gains of investments.

Decay Applications may include radioactive decay, radiocarbon dating, and cooling.

One common term for decay applications is half-life. i.e. At a certain time, say t*,

half of the original amount, C, remains. i.e. y(t*) / C = 1/2

Here are some properties of logarithms and exponentials that you may find useful.

log_a(x) = ln (x) / ln(a) , ln e^x = x , e ^ln^x = x , exp [ln(ab)] = ab

log_a (bc) = log_a (b) + log_a (c) , log_a (b/c) = log_a (b) ˗ log_a (c)

e^ab = (e^a)^b , ln (e^ab) = ln ((e^a)^b) = ab , e^{(ln a)b} = [e^{(ln a)}]^b = a^b

a^xy = (a^x) ^y , (ab)^x = a^x b^x , exp (ln a^r)t = exp [ln (a^r)^t] = a^rt

In some applications the growth or decay of the function, y(t), may take the form

y(t) = C a ^{r t}

i.e. If k = ln (a ^r ) from (2) y(t) = C exp { ln [a ^r )] t } = C (a ^{r t} ) = C a ^{r t}