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
Doubling time can be calculated given any exponential equation in the form . Given this equation, there exists an equation of the form that is equivalent to . Since the base of the second equation is 2, it doubles every time c increases by 1. So c is the doubling time.
|1||Since y=y, the right hand sides of the two equations are equal to each other.|
|2||Divide both sides by a, eliminating it from the equation. A consequence of this step is that a!=0.|
|3||Use the definition of logarithm to transform the equation.|
|4||Use the power rule of logarithms to pull cx out of the logarithm.|
|5||Divide both sides by x, eliminating it from the equation. A consequence of this step is that x!=0.|
|6||Transform the log base d to the natural log using the change of base formula.|
|7||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.|
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