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.

Example:

Two raised to the third power.

The expression 23 can be represented by the tree to the left. In this example, the base is two, so at each level of the tree the node will split into two. If we draw the tree and count the number of leaves then we quickly come up with the answer eight: .

Example:

Three raised to the second power.

The expression 32 can be represented by the tree to the left. In this example, the base is three, so at each level of the tree the node will split into three. If we draw the tree and count the number of leaves then we quickly come up with the answer nine: .

Logarithms

The above interactive 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.

Example:

Three raised to the second power.

The expression can be represented by the tree to the left. Note that the base is two, which corresponds to a branching factor of two. Instead of counting the number of leaves, we count the number of levels in the tree.

Example:

Three raised to the second power.

The expression can be visualzed by the tree to the left. Note that the base is three, which corresponds to a branching factor of three. To evaluate this expression we count the number of levels in the tree: . Another way to think of this is how many times do we divide nine by three and end up with one?

Conclusion

Try and formulate some simple problems, such as 42 or log3 27, and draw 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.