The exponential growth the term b>1
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
for example,
Example: 2 months ago you had 3 mice, you now have 18. 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
Swap sides: 2k = ln(6)
Divide by 2: k = ln(6)/2
Notes:
The step where we used ln(ex)=x is explained at Exponents and Logarithms.
we could calculate k ≈ 0.896, but it is best to keep it as k = ln(6)/2 until we do our final calculations.
We can now put k = ln(6)/2 into our formula from before:
y(t) = 3 e(ln(6)/2)t
Now let's calculate the population in 2 more months (at t=4 months):
y(4) = 3 e(ln(6)/2)×4 = 108
And in 1 year from now (t=14 months):
y(14) = 3 e(ln(6)/2)×14 = 839,808
As, per the formula for exponential growth the factor b will tells whether there is exponential growth or decay.
so, when b>1 then it is exponential growth and b<1 shows decays.
Learn more about exponential growth here:
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