Exponential Relationship

A verbal description of the relationship

The exponential relationship can be put in two contexts, first as some number to a variable exponent. For example, 2^8 power is an exponential representation of eight population doublings. Another context is that the exponential function can be used to represent the growth or decay of a quantity with time that increases (or decreases) relative to the current amount. For example, the growth of money in a bank acount that is getting interest and compounded very frequently will be described with an exponential curve. The reason for this is that integration of the instantaneous increase over time gives an exponential function.

The equation and the meaning of x, y, and other parameters

Y = m*10^x or Y=m*e^x

x is the exponent to the numbers 10 or e

m is the often the "initial" value when X=0

One or more examples

Plot the increase in the population of rabbits that are growing at an instantaneous growth rate of 1 new rabbit per adult rabbit per month. Start with 100 rabbits.

Y = 100 * e ^(1 * X)

X is the time in months

100 is the initial number of rabbits.

e is 2.7128...

Dimensional analysis:

The instantaneous rate is in units if t^-1 (1/t), when this is multiplied by the time, the value is dimensionless. The exponent has to be dimensionless. This value is multiplied by the initital value of 100 rabbits to get, Y, the rabbits at any time X.

A graph

Algebraic rules that apply to the use of this equation

There are rules that relate to exponents.

The exponent is calculated before other calculations. For example 2*10^2 = 200, not 400.

When multiplying numbers that are represented in an exponential form you add the exponents. For example (10^2) * (10^3) = 10^5.

Taking an exponent of another term that has an exponent is the same as multiplying the exponents. For example (and this is an important example) if 10 = e^2.303 then 10^2 = e^(2.303*2). This rule tells you how to convert between terms with different root values for exponents.

Characteristic values

for the equation Y=m*e^X

at X = 0, e^X =1 and thus Y =m

Y cannot = 0

as X increases Y increases if e is taken to a positive power, but it can decrease toward 0 if the equation includes a term Y=m*e^(-1*X)