The 5 functions you need to know

Describe scalability, growth and income more precisely

Three exponential functions — green has base 4, blue has base 3, red has base 2. Image by Martin Thoma

Knowing the following set of functions helps you to describe the change in income, the growth of your company, the number of COVID-19 cases, …

Logarithmic Functions

Logarithmic functions are also monotonically increasing, but after a quick initial phase they are only increasing very slowly:

Image by Martin Thoma.

Logarithms have a base. In the example above you can see two examples with base 2 and one example with base 10. The higher the base, the slower the function grows. Logarithmic functions have two phases:

• 0 < x < 1: Negative phase. The closer to get to zero, the more negative the logarithms value becomes.
• x = 1: The logarithm of 1 is zero for all bases.
• x > 1: The logarithm keeps growing.

One invariant to know is

This means the logarithm to base b reverses the exponential function to base b.

In computer science, this kind of growth behavior is associated with trees: