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Exponential Decay

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Tutorial

- Exponential Decay Formula
- Calculating the Amount of a Substance
- Solving for Decay Time

**Exponential Decay Formula**

The general exponential equation is y = a • b^{x}, where a base number, b, is raised to a variable power, x. There is also a scalar multiplier, a, in front of the exponential expression.

Exponential decay can be modeled with a similar equation:

With exponential decay, b represents the rate of decay, (expressed as a decimal), and is subtracted from 1 to represent decay. We can call (1 – b) the decay factor, since it is multiplied by a (the initial value) x number of times.

**Calculating the Amount of a Substance**

Many of us take aspirin or ibuprofen when we have a headache, fever, or other ailment. A certain amount of active ingredients enter our bloodstream, but are consumed by our bodies over time, such that only a portion of the drug is still in our blood stream. Suppose we take an 80 mg pill of aspirin, which dissolves at a rate of 60% every hour. How many milligrams is in the blood stream after 4 hours?

From the scenario, we can draw the following values for certain variables in the exponential decay formula:

- a = 80 mg
- b = 0.6 (60% expressed as a decimal)
- x = 4 (hours)

After 4 hours, only 2.048 mg of the medicine is in the blood stream. Perhaps this is why doctors recommend taking a new dosage every 4 hours.

**Solving for Decay Time**

How long do you think a world wide Rock Paper Scissors contest would take, if all 7 billion humans on earth entered the tournament? We can answer this question using our exponential decay model.

The first round would have 7 billion contenders, with half of them being eliminated each round. This means that our rate of decay is 50%, since only 50% of the population remains in the tournament each round. This must continue until we have only 1 person left as the world champion in Rock Paper Scissors.

We can now identify knowns and unknowns to use in our formula:

- y = 1 (the world champion)
- a = 7,000,000,000 (total number of players)
- b = 0.5 (50% as a decimal)
- x = unknown, number of rounds to crown a champion

Since x represents the number of rounds in the tournament, we will round up to the nearest whole number. This means the entire world can play a full Rock Paper Scissors tournament in only 33 quick rounds.

There are a couple of things to note when solving these types of problems:

- Before doing anything about the variable exponent, divide the equation by the a-value. This is because the exponent is attached to the expression inside of the parentheses (1 – b), and not a.
- You can apply the log of any base to undo the exponent, so long as the base is the same on both sides of the equation. Most often, we use common log (base 10) or natural log (base e).
- In applying the log to both sides, we use the power property of logarithms to bring the exponent outside of the log function, and place it as a scalar multiplier in front of the log. This allows us to isolate the variable through division.

Formulas to Know

- Exponential Decay