Exponential Growth

Do you remember the beginning of the pandemic, when exponential growth was all the talk?

Let’s revisit this topic for a moment and learn a cute little trick, how to quickly estimate the exponential growth of a quantity without having to use a calculator.

You have all heard of compound interest, right?

You invest a capital (or take a loan) at an annual interest rate of p%.

Say, you invest 100€ at a rate of 5%

The first year, you get 5€ interest and you have 105€.
The next year, you get 5.25€, so you have 110.25€.

After how many years (k), do you have 200€?

This is the promised cute Math trick, to quickly get a good estimate:

k ≈ 69/p

In our example, with p = 5, we get k ≈ 14

In fact, after 14 years, you have 198€.

How does this work?

First, let’s look at the exact solution:

(1 + ᵖ/₁₀₀)ᵏ = 2
⇔
log[(1 + ᵖ/₁₀₀)ᵏ] = log(2)
⇔
k·log(1 + ᵖ/₁₀₀) = log(2)
⇔
k = log(2)/log(1 + ᵖ/₁₀₀)

The logarithm is a smooth function, which means, it has several derivatives (remember calculus from school?).

For such functions, there exists a theorem (Taylor’s formula), which says for an arbitrary number ε:

f(x + ε) = f(x) + ε·f'(x) + δ(ε)

f’ is the derivative of f

where δ(ε) is called the error.

The smaller, ε gets, the smaller the error will be.

So, for small ε we can estimate:

f(x + ε) ≈ f(x) + ε·f'(x)

In our case, f(x) = log(x) and f'(x) = 1/x

We set x = 1 and ε = ᵖ/₁₀₀ and get with log(1) = 0

log(1 + ᵖ/₁₀₀) ≈
log(1) + ᵖ/₁₀₀ ·1/1 = 0 + ᵖ/₁₀₀ = ᵖ/₁₀₀

We can look up log(2) ≈ 0.69 and get with our formula from the previous toot:

k ≈ 0.69/(ᵖ/₁₀₀) = 69/p

What we wanted to show!

Neat, right?

As we all carry scientific calculators with us all the time, or we can even ask the search machine of our choice for an exact solution too every such trivial problem, estimates like this one, are not really necessary.

Still, I believe they are useful to get a feeling for numbers and functions (in this case, the exponential function).

So, I consider exercises like this not to be completely in vain.

Besides: This a form of art and game, which are two things that make us human.

Remarks

In school, you may have learned that log(x) is the logarithm of x to the base of 10, whilst ln(x) is the natural logarithm to the base of Euler’s number e.

Mathematicians don’t use the logarithm to the base of 10, so for us, log(x) is the natural logarithm.

You might have wondered, why I wrote the entire text in plural, using the pronoun we.

This is also a Mathematicians’ convention. We don’t do that, because we consider ourselves to be majestic, but because the author is inviting the reader to join them on their journey.