Logarithms Exponents and Trees

Base

Logarithms and exponents have a base which is a number. Similarly, as a tree grows the tree will sprout new branches. This “branching factor” can be thought of the base of a tree, exponent, or logarithm. Take for example trees of base two, three and four.

At each level of the tree, new branches sprout equal to the base. If you were to try and draw trees with higher bases, you might be surprised by how quickly they grow. This consequence of branching is one of the fundamental ideas that separates exponents and logarithms from “linear thinking”. Instead, exponents are like thinking in a “multiplicative-way”. At each level the number of nodes is multiplied instead of having some constant added. Inversely, taking the logarithm of a number is like thinking in a “division-like-way”.

Trees

Play around with the variables below to see how the number of leaves change based on the number of levels in the tree and the branching factor at each level.

Exponents

The above image describes the mathematical form of exponentiation. The base corresponds to the branching factor, y corresponds with the number of levels, and finally x corresponds to the number of leaves in the tree.

Logarithms

The above image also describes the mathematical form of taking the logarithm of a number, which is the inverse operation of exponentiation. Here the roles are reversed, instead of solving for the number of leaves, we solve for the number of levels in the tree.

Conclusion

Try and formulate some simple problems, such as: 25 or log3 27, and solve them by drawing a little tree to solve them. Hopefully, this gives you a simple tool to visualize and reason about problems concerning exponents and logarithms. Maybe the next time you look at a tree, bush, or plant, you will see it a little bit more like a mathematician.