Doubling Time

Pronunciation: /ˈdʌ blɪŋ taɪm/ Explain

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How does the doubling time change when a changes? Why? How does the doubling time change when b changes? Why?
Manipulative 1 - Doubling Time Created with GeoGebra.

Doubling time is the amount of time it takes for an exponential function to double. If an exponential function goes from 1 to 2 in 10 seconds, it will go from 2 to 4 in 10 seconds and 4 to 8 in 10 seconds. An exponential function always doubles in the same amount of time.

Manipulative 1 illustrates doubling time for an exponential function of the form
f(x)=a*e^(b*x).

Discovery

  1. How does the doubling time change when a changes? Why?
  2. How does the doubling time change when b changes? Why?

Rule of 72

The rule of 72 can be used to approximate doubling time of investments at a given interest rate. The rule of 72 states:
t=0.72/r
where r is the interest rate.

Calculating Doubling Time

Doubling time can be calculated given any exponential equation in the form y=a*d^(b*x). Given this equation, there exists an equation of the form y=a*2^(c*x) that is equivalent to y=a*d^(b*x). Since the base of the second equation is 2, it doubles every time c increases by 1. So c is the doubling time.

StepEquationDescription
1a*d^(bx)=a*2^(cx)Since y=y, the right hand sides of the two equations are equal to each other.
2d^(bx)=2^(cx)Divide both sides by a, eliminating it from the equation. A consequence of this step is that a!=0.
3log base d(2^(cx))=bxUse the definition of logarithm to transform the equation.
4cx*log base d(2)=bxUse the power rule of logarithms to pull cx out of the logarithm.
5c*log base d(2)=bDivide both sides by x, eliminating it from the equation. A consequence of this step is that x!=0.
6c*ln(2)/ln(d)=bTransform the log base d to the natural log using the change of base formula.
7c=b*ln(d)/ln(2)Divide both sides by the logarithmic ratio. c is now on one side of the equation by itself. What is on the right hand side of the equation is the doubling time.
Table 1: Derivation of doubling time.

References

  1. Rule of 72. investopedia.com. Investopedia. Forbes Digital. Last Accessed 8/6/2018. http://investopedia.com/terms/r/ruleof72.asp.
  2. Lodge, Sir Oliver. Easy Mathematics; or, Arithmetic and Algebra for General Readers. pp 353-356. www.archive.org. Macmillan and Company. 1910. Last Accessed 8/6/2018. http://www.archive.org/stream/easymathematicso00lodguoft#page/353/mode/1up. Buy the book
  3. Math 120 Book. pp 698-699. www.archive.org. Last Accessed 8/6/2018. http://www.archive.org/stream/Math120Book-Entire/Math120-fullBook#page/n334/mode/1up/search/doubling.

Cite this article as:

McAdams, David E. Doubling Time. 7/4/2018. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/d/doublingtime.html.

Image Credits

Revision History

7/4/2018: Removed broken links, updated license, implemented new markup, implemented new Geogebra protocol. (McAdams, David E.)
12/16/2008: Initial version. (McAdams, David E.)

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