Editing Exponential growth (section)

==Basic formula==
[[File:Exponentielles wachstum2.svg|thumb|exponential growth:<br/> <math>\begin{align} a&=3 \\ b&=2 \\ r&=5 \end{align}</math>]]
[[File:Exponentieller zerfall2.svg|thumb|exponential decay:<br/> <math>\begin{align} a&=24 \\ b&=\frac{1}{2} \\ r&=5\end{align}</math>]]
A quantity {{mvar|x}} depends exponentially on time {{mvar|t}} if
<math display="block">x(t)=a\cdot b^{t/\tau}</math>
where the constant {{math|''a''}} is the initial value of {{mvar|x}}, <math display="block">x(0) = a \, ,</math> the constant {{math|''b''}} is a positive growth factor, and {{math|''τ''}} is the [[time constant]]—the time required for {{mvar|x}} to increase by one factor of {{math|''b''}}:
<math display="block">x(t+\tau) = a \cdot b^{(t+\tau)/\tau} = a \cdot b^{t/\tau} \cdot b^{\tau/\tau} = x(t) \cdot b\, .</math>

If {{math|''τ'' > 0}} and {{math|''b'' > 1}}, then {{mvar|x}} has exponential growth. If {{math|''τ'' < 0}} and {{math|''b'' > 1}}, or {{math|''τ'' > 0}} and {{math|0 < ''b'' < 1}}, then {{mvar|x}} has [[exponential decay]].

Example: ''If a species of bacteria doubles every ten minutes, starting out with only one bacterium, how many bacteria would be present after one hour?''  The question implies {{math|1=''a'' = 1}}, {{math|1=''b'' = 2}} and {{math|1=''τ'' = 10 min}}.

<math display="block">x(t)=a\cdot b^{t/\tau} = 1 \cdot 2^{t/(10\text{ min})}</math>
<math display="block">x(1\text{ hr}) = 1\cdot 2^{(60\text{ min})/(10\text{ min})} = 1 \cdot 2^6 =64.</math>

After one hour, or six ten-minute intervals, there would be sixty-four bacteria.

Many pairs {{math|(''b'', ''τ'')}} of a [[dimensionless]] non-negative number {{math|''b''}} and an amount of time {{math|''τ''}} (a [[physical quantity]] which can be expressed as the product of a number of units and a unit of time) represent the same growth rate, with {{math|''τ''}} proportional to {{math|log ''b''}}. For any fixed {{math|''b''}} not equal to 1 (e.g. ''[[E (mathematical constant)|e]]'' or 2), the growth rate is given by the non-zero time {{math|''τ''}}. For any non-zero time {{math|''τ''}} the growth rate is given by the dimensionless positive number&nbsp;{{math|''b''}}.

Thus the law of exponential growth can be written in different but mathematically equivalent forms, by using a different [[exponentiation|base]]. The most common forms are the following:
<math display="block">x(t) = x_0\cdot e^{kt} = x_0\cdot e^{t/\tau} = x_0 \cdot 2^{t/T}
= x_0\cdot \left( 1 + \frac{r}{100} \right)^{t/p},</math>
where {{math|''x''<sub>0</sub>}} expresses the initial quantity {{math|''x''(0)}}.

Parameters (negative in the case of exponential decay):
* The ''growth constant'' {{math|''k''}} is the [[frequency]] (number of times per unit time) of growing by a factor {{math|''e''}}; in finance it is also called the logarithmic return, [[continuous compounding|continuously compounded return]], or [[Compound interest#Force of interest|force of interest]].
* The ''[[e-folding|e-folding time]]'' ''τ'' is the time it takes to grow by a factor ''[[E (mathematical constant)|e]]''.
* The ''[[doubling time]]'' ''T'' is the time it takes to double.
* The percent increase {{math|''r''}} (a dimensionless number) in a period {{math|''p''}}.
The quantities {{math|''k''}}, {{math|''τ''}}, and {{math|''T''}}, and for a given {{math|''p''}} also {{math|''r''}}, have a one-to-one connection given by the following equation (which can be derived by taking the natural logarithm of the above):
<math display="block">k = \frac{1}{\tau} = \frac{\ln 2}{T} = \frac{\ln \left( 1 + \frac{r}{100} \right)}{p}</math>
where {{math|1=''k'' = 0}} corresponds to {{math|1=''r'' = 0}} and to {{math|''τ''}} and {{math|''T''}} being infinite.

If {{math|''p''}} is the unit of time the quotient {{math|''t''/''p''}} is simply the number of units of time. Using the notation {{mvar|t}} for the (dimensionless) number of units of time rather than the time itself, {{math|''t''/''p''}} can be replaced by {{mvar|t}}, but for uniformity this has been avoided here. In this case the division by {{math|''p''}} in the last formula is not a numerical division either, but converts a dimensionless number to the correct quantity including unit.

A popular approximated method for calculating the doubling time from the growth rate is the [[rule of 70]],
that is, <math>T \simeq 70 / r</math>.

{{wide image|doubling_time_vs_half_life.svg|640px|Graphs comparing doubling times and half lives of exponential growths (bold lines) and decay (faint lines), and their 70/''t'' and 72/''t'' approximations. In the [http://upload.wikimedia.org/wikipedia/commons/8/88/Doubling_time_vs_half_life.svg SVG version], hover over a graph to highlight it and its complement.}}