Exponential Growth and Decay

Exponential growth can be amazing!

 

Let us say we have this special tree.

It grows exponentially, following this formula (e is Euler's number):

 

Height (in mm) = ex

e^x

 

 

No tree could ever grow that tall!
So when people say "it grows exponentially" ... just think what that means.

Growth and Decay

But sometimes things can grow (or the opposite: decay) exponentially, at least for a while.

So we have a generally useful formula:

y(t) = a × ekt

Where y(t) = value at time "t"
a = value at the start
k = rate of growth (when >0) or decay (when <0)
t = time

 

Example: 2 months ago you had 3 mice, you now have 18.

Mice

Assuming the growth continues like that

  • What is the "k" value?
  • How many mice 2 Months from now?
  • How many mice 1 Year from now?

 

Start with the formula:

y(t) = a × ekt

We know a=3 mice, t=2 months, and right now y(2)=18 mice:

18 = 3 × e2k

Now some algebra to solve for k:

Divide both sides by 3:   6 = e2k
     
Take the natural logarithm of both sides:   ln(6) = ln(e2k)
     
ln(ex)=x, so:   ln(6) = 2k
     
Rearrange:   k = ln(6)/2

(Note: k ≈ 0.896, but it is best to keep it as ln(6)/2 until we do our final calculations.)

Now, we want to know the population in 2 more months (at t=4 months), and in 1 year from now (t=14 months):

y(4) = 3 e(ln(6)/2)×4 = 108

y(14) = 3 e(ln(6)/2)×14 = 839,808

 

That's a lot of mice! I hope you will be feeding them properly.

Exponential Decay

Some things "decay" (get smaller) exponentially.

Example: Atmospheric pressure (the pressure of air around you) decreases as you go higher.

It decreases about 12% for every 1000 m: an exponential decay.

The pressure at sea level is about 1013 hPa (depending on weather).

 

 

Start with the formula:

y(t) = a × ekt

We know

So:

891.44 = 1013 ek×1000

Now some algebra to solve for k:

Divide both sides by 1013:   0.88 = e1000k
     
Take the natural logarithm of both sides:   ln(0.88) = ln(e1000k)
     
ln(ex)=x, so:   ln(0.88) = 1000k
     
Rearrange:   k = ln(0.88)/1000

 

Now we know "k" we can write:

y(t) = 1013 e(ln(0.88)/1000)×t

 

And finally we can calculate the pressure at 381 m, and at 8848 m:

 

y(381) = 1013 e(ln(0.88)/1000)×381 = 965 hPa

y(8848) = 1013 e(ln(0.88)/1000)×8848 = 327 hPa

 

(In fact pressures at Mount Everest are around 337 hPa ... not bad!)