Exponential growth and decay models

See also: Exponential Growth and Decay

The Natural Functions

The Natural Exponent Function

The base e is a transcendental function defined by an infinite series, which approximately is 2.71828:

$e=\sum _{n=0}^{\infty }{\frac {1}{n!}}={\frac {1}{0!}}+{\frac {1}{1!}}+{\frac {1}{2!}}+{\frac {1}{3!}}+{\frac {1}{4!}}+\cdots$.

e is one of the most important number in mathematics because of its use in most natural growth processes calculations. Since it is an exponential function it follows all the laws of exponential functions. The natural exponent is the inverse function of natural logarithm, so that $e^{\ln x}=x\,$. On the right is the graph of $e^{x}\,$.

The Natural Logarithm

The natural log is a logarithmic function with the base e: $\log _{e}x\,$. It has a special notation of $\ln x\,$. Since the natural logarithm is a logarithmic function it follows all the laws of logarithmic functions. The natural logarithm is the inverse function of natural exponent, so that $\ln e^{x}=x\,$. On the right is the graph of $\ln x\,$.

Exponential growth and decay

One of the primary usage of the natural functions is the calculation of natural growth or decay, which can be calculated using the following formula: $y\left(t\right)=y_{0}e^{kt}\,$, where y(t) is the final value, $y_{0}$ is the initial value, k is the growth constant, t is the elapsed time. One special decay is the half life of an element there k is defined as: $k=-{\frac {\ln 2}{\mbox{half-life}}}$. Exponential growth and decay have a very broad range of applications from measuring the growth bacteria colony to calculating involving interest.

Example of Exponential Growth

A bacterial colony was started with 200 bacteria, in 2 hours the colony has grown to 600 bacteria. Predict the quantity of bacteria in the colony in 6 hours?

From this we know:

• y(t) = 600
• $y_{0}$ = 200
• k = x
• t = 2

Now we substitute the values into the formula:

$600=200e^{2k}\,$

Now we ensure that e is alone:

${\frac {600}{200}}=e^{2x}\,$

Since ln is the inverse function we use it to remove the base e.

$\ln 3=\ln e^{2x}\,$

We can now solve for x.

$1.1\approx 2x\,$, so the growth factor for the colony is approximately .55

Now we can predict the quantity of bacteria after 6 hours.

$x=200e^{6\times .55}\,$

$x=5423\ bacteria\,$

Example of Half-life decay

The element Carbon-14 has a half-life of 5730 years. When will a 100 gram sample of C-14 be reduced to 20% C-14 and 80% C-12?

• y(t) = 20% * 100 = 20
• $y_{0}$ = 100
• half-life = 5730
• t = x

We substitute all knows into the function. Note the special formula for the growth constant.

$20=100e^{\left(-{\frac {\ln 2}{5730}}\times t\right)}$

Now we ensure e is alone:

$.2=e^{\left(-{\frac {\ln 2}{5730}}\times t\right)}$

Since ln is the inverse function we use it to remove the base e.

$\ln .2=\ln e^{\left(-{\frac {\ln 2}{5730}}\times t\right)}$

$\ln .2=-{\frac {\ln 2}{5730}}\times t$

Now we rearrange the equation to isolate t.

$t=-5730{\frac {\ln .2}{\ln 2}}$

Solving for t we get that the sample will be reduced to 20% C-14 and 80% C-12 after about 13304.6 years.

$t\approx 13304.6\ years$

Resources

• Exponential growth and decay models. Written notes created by Professor Esparza, UTSA.
• Exponential growth and decay models Continued. Written notes created by Professor Esparza, UTSA.

Licensing

Content obtained and/or adapted from:

• Special Functions and Transformations, Wikibooks: A-level Mathematics under a CC BY-SA license