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An exponential function is a function whose rate of growth is controlled by the independent variable. The general form of an exponential function is: $$y=ab^x$$ where $b$ is the base, $x$ is the independent variable or exponent, and $y$ is the dependent variable. $a$ is some non-zero constant.

A function increases exponentially when the base is greater than $1$. A function decreases exponentially when the base is between $0$ and $1$. A function is constant when the base is exactly $0$ or $1$. Tangentially, a negative base causes the function to be undefined at fractional inputs, and to oscillate between positive and negative outputs at integer value inputs.

Exponential decay describes the functions that decrease exponentially (as opposed to exponential growth, where the functions increase exponentially). In exponential decay, the dependent variable decreases as the independent variable increases, while both variables increase in exponential growth.

Here's a picture of the difference between the two kinds of exponential functions:

The decrease occurs in exponential decay because $b$ is a positive decimal less than $1$, so multiplying it by itself $x$ many times will decrease $y$.

$$\color{purple}{\text{Expiinium}}$$

=== $$\text{How to calculate rate of decay, half life, and more!}$$

Congratulations! You've discovered a new element, and you've decided to name it after $\color{purple}{\text{your favorite website}}$!

$\color{purple}{\text{Expiinium}}$ decays exponentially. You start with some amount of the element, and after two hours you notice that you have $12$ ounces left. Then, after two more hours, there's only $6.75$ ounces left!

Understanding the formula

Since we know $\color{purple}{\text{Expiinium}}$ decays exponentially, we know that the same proportion of the element will disappear with every unit of time. This means we can use the formula for exponential decay: $$y=a(1-r)^t$$ Where $a$ is the initial amount of $\color{purple}{\text{Expiinium}}$, $y$ is the final amount of $\color{purple}{\text{Expiinium}}$, $r$ is the rate of decay of the element (that is, how much of the element decays with every unit of time), and $t$ is the amount of time that passes.

And this makes sense! In our problem, $t$ is measured in hours. The value $(1-r)$ is always less than one, so multiplying the initial amount of $\color{purple}{\text{Expiinium}}$ by $(1-r)^1$ exactly tells us how much $\color{purple}{\text{Expiinium}}$ is left over after $1$ hour. Multiplying this new amount by $(1-r)$ again tells us how much $\color{purple}{\text{Expiinium}}$ is left over after two hours, and so on!

How to find rate of decay

One of the best parts of the exponential decay formula is that you can choose any initial and final amounts, and the rate will stay constant regardless.

Let's call our $12$ ounces of $\color{purple}{\text{Expiinium}}$ our initial amount and our $6.75$ ounces of $\color{purple}{\text{Expiinium}}$ our final amount. We know that two hours passed between these two measurements, so $t=2$. Let's plug these values into our formula: $$y=a(1-r)^t$$ $$6.75=12(1-r)^2$$ Now all we have to do is solve for $r$: $$6.75=12(1-r)^2$$ $$(1-r)^2=\frac{6.75}{12}$$ $$(1-r)^2=0.5625$$ $$1-r=\sqrt{0.5625}$$ $$1-r=0.75$$ $$r=0.25$$

By definition, $r$ is the amount of material that decays per unit of time; in this problem, we see that $.25=25\%$ of $\color{purple}{\text{Expiinium}}$ decays per hour (one unit of time).

How to find initial amount

Let's suppose you want to find how much $\color{purple}{\text{Expiinium}}$ you had in the first place!

Well, we can use the exponential decay formula again. We know that after two hours, our initial value decayed to $12$ ounces*. Then our initial amount is unknown, our final amount is $12$, our rate of decay is constant (so it's still $0.25$), and our time is $2$ hours: $$y=a(1-r)^t$$ $$12=a(1-0.25)^2$$

*We could use any known value for this, so $y=6.75, t=4$ works as well!

There were just over $21$ ounces of $\color{purple}{\text{Expiinium}}$ at the start. Wow! We can get this value by solving for $a$ in the equation above: $$12=a(1-0.25)^2$$ $$12=a(0.75)^2$$ $$12=a(0.5625)$$ $$a=\frac{12}{0.5625}$$ $$a=21\frac{1}{3}$$ And this makes sense! If we start with $21\frac{1}{3}$ ounces of $\color{purple}{\text{Expiinium}}$, $25\%$ of it will decay within an hour, so there's only $75\%$ of it left after an hour: $$21\frac{1}{3}\cdot 0.75=16$$ So there's $16$ ounces left after an hour. To find how much is left after one more hour ($2$ hours total), we perform the calculation again: $$16\cdot 0.75=12$$ And that's the $12$ ounces we have after two hours!

How to find half-life

Half-life is the amount of time it takes for exactly half of the material to decay; that is, if you have $2x$ ounces of $\color{purple}{\text{Expiinium}}$, the half-life is the time until there's only $x$ ounces of $\color{purple}{\text{Expiinium}}$ left.

Thus, since we know our initial value ($2x$), our final value ($x$), and our rate of decay ($0.25$, as above), can use our exponential decay formula to solve for time! $$y=a(1-r)^t$$ $$x=2x(1-0.25)^t$$

We can find the half-life by solving for $t$ in the formula above. Note how the two $x$s cancel out: $$x=2x(1-.0.25)^t$$ $$x=2x(0.75)^t$$ $$\frac{x}{2x}=(0.75)^t$$ $$0.5=(0.75)^t$$ At this point, we have to use logarithms to solve for $t$: $$t=\log_{0.75}0.5$$ $$t\approx 2.41$$ Thus, it takes $2.41$ hours (or approximately $2$ hours and $24.6$ minutes) for half of the $\color{purple}{\text{Expiinium}}$ to decay.

We can check our work to make sure we have the right number. Suppose we start with $10$ ounces of $\color{purple}{\text{Expiinium}}$; we want there to be $\frac{10}{2}=5$ ounces of $\color{purple}{\text{Expiinium}}$ left after $2.41$ hours: $$y=a(1-r)^t$$ $$y=10(1-0.25)^{2.41}$$ $$y=10(0.75)^{2.41}$$ $$y=10(0.4999)$$ $$y\approx 5$$ So our math is correct!

(Video) Exponential Decay and Finding Half-Life

by PatrickJMT

Here's a video by PatrickJMT showing you a typical exponential decay problem involving half-life.

$\large{\text{Summary}}$

"The half-life is the amount of time required for the substance to decay to half of its original amount." For these problems we use the equation $y=a(1+r)^t$ where $r$ is the annual growth rate. So for decay, all you do is subtract the rate.

Exponential Decay

We see exponential decay in a lot of aspects of life, including radioactive decay.

The formula to calculate exponential decay is:

$$y(t)=y_0e^(- \lambda t)$$

, where $t$ is time, $\lambda$ is the decay constant, and $y_0$ is the initial volume.

In the next lessons, you'll learn how to solve exponential equations and graphing exponential equations.