EXPONENTS & LOGARITHMS | TUTORIAL | MATH

Exponents and Logarithms: A Beginner’s Starter Guide to Understanding Complex Math

Unlock the secrets of growth, scaling, and problem solving with these fundamental — and simple — mathematical concepts.

6 min readNov 24, 2024

Photo by Dan Cristian Pădureț on Unsplash

Did you know that, if you doubled two pennies every single day, by day 19 you would reach 1,048,576 pennies, or $10,485.76? And by the end of the month, you would have reached $10,737,418.24!

Last time, we covered the basic operations of addition, subtraction, multiplication and division — if you haven’t seen it, here’s a link!

Today, we’ll dive into a slightly more complex part of math: exponentiation, and its inverse, the logarithm.

Whether you’re a programmer tackling time complexity or just want to know more about this field, exponents and logarithms are two essential concepts you need to master.

Understanding Exponentiation

What Are Exponents?

An exponent is simply a shorthand way of writing down a repeated multiplication. For example, 2³ means 2 × 2 × 2 = 8.

The big number in the center is called the base, and the little number in the top-right is called the exponent, or sometimes ‘power’.

In the 2³ example, we can say either “2 to the power of 3” or “2 raised to the 3rd power”.

Do Not Confuse Exponents with Factors

A lot of people take the little number and think that you have to multiply the two numbers together.

So, for example, 2³ = 6. This is wrong.

Make sure you understand this: 2³ does not mean 2 × 3, it means 2 × 2 × 2.

Some Powers Have Special Names

Any number x raised to the power of 2 gets the name of a square.

For example, 2² can be read as 2 squared, and the result, 4, is the square of 2.

Any number x raised to the power of 3 gets the name of a cube.

For example, 2³ can be read as 2 cubed, and the result, 8, is the cube of 2.

Why Exponents Matter

Exponents help us understand growth, from calculating interest on a loan to tracking viral trends.

Each increase corresponds to a multiplication rather than an addition, leading to rapid growth.

The Key Rules of Exponentiation

Exponentiation has a few fundamental rules that we need to keep in mind.

Product rule: aⁿ × aᵐ = aⁿ⁺ᵐ (e.g., 2³ × 2² = 2⁵ = 32).

Remember that an exponent simply represents repeated multiplication. So, with 2³ × 2² = 2⁵ it is because we have two distinct groups that can be separated as (2 × 2 × 2) × (2 × 2) = 2⁵.

Power of a power: (aⁿ)ᵐ = aⁿ*ᵐ (e.g., (2²)³ = 2⁶ = 64).

This is because we first calculate inside the parentheses and then use the result of that as the base for the exponent outside of the parentheses.
(2²)³ = 4³ = 4 × 4 × 4 = 64

Negative exponents: a⁻ⁿ = 1 / aⁿ (e.g., 2⁻² = 1 / 2² = 1/4).

Fractional exponents: a^(1/n) = nth root of a (e.g., 8^(1/3) = the cube root of 8 = 2).

Introducing Logarithms: The Inverse Operation of Exponentiation

What Are Logarithms?

Exponents tell us how many times to multiply a base by itself.

Logarithms do the opposite. They ask, “What exponent produces my content?”

For instance, log₂(8) = 3 because 2³ = 8.

If bⁿ = c, then log_b(c) = n.

b is the base
c is the content of the logarithm
n is the exponent

Logarithms essentially ask, “What power do I have to raise the base to if I want to get the content as a result?”.

Types of Logarithms

There are different types of logarithms:

Common logarithms use a base of 10.

They are very widely used, which is where the name came from.
In some parts like here in Romania, they are simply written as lg instead of log. In others, it’s common to simply write log without a base.

Natural logarithms use Euler’s number as a base (written with an e, ~2.71828).

These are also commonly used. They are simply written with ln instead of log.

Other logarithms use the notation log_b(c) = n.

How Logarithms Are Useful

From measuring sound intensity — or volume — with decibels to assessing an earthquake’s strength with the now defunct Richter scale, logarithms are everywhere when you try to look for them.

They are also useful in computer science for problem solving, helping with time complexity, or solving binary trees and making algorithms.

Logarithms Also Have Some Properties of Their Own

Logarithms, like all other operations, have certain properties and rules they need to abide by.

Logarithms have 3 existential conditions. log_a(b) = x ONLY IF:

a is strictly greater than 0,
a is not equal to 1, and
b is strictly greater than 0.

The above can also be written in mathematical notation as:
∃log_a(b) ⇔ {a > 0; a ≠ 1; b > 0}

If any of the three conditions are not met, then, the logarithm isn’t valid and cannot exist.

Now for logarithm properties — i.e. some tips for solving problems — there are some more:

If both the base a and the content b are equal, then the result is always 1. log_a(a) = 1 because any number raised to the power of 1 equals that number (essentially multiplying a number by 1).
If the content is 1 then, no matter the base, the result is 0. log_a(1) = 0 because any number raised to the power of 0 equals 1.
The formula for changing the base of a logarithm is: log_a(b) = log_x(b) / log_x(a)
log_a(b^c) = c × log_a(b)
log_a(a^c) = c because log_a(a) cancels itself out.
log_a(b) + log_a(c) = log_a(b × c)
log_a(b) — log_a(c) = log_a(b / c)

How Exponents and Logarithms Work Together

Exponents and logarithms are two sides of the same coin.

For example, log₂(2ⁿ) = n and 2^(log₂(n)) = n.

For a practical example, you can think of a population of 2 which reproduces at a rate of 2ⁿ. In other words, each generation will be twice as big as the preceding generation.

With this in mind, which generation would have 1024 members? Well, it’s as simple as taking a logarithm of base 2 with a content of 1024, log₂(1024).

Now, the question is, what power of 2 gives 1024? Well, in case you haven’t seen my article on understanding binary, we can simply multiply 2 by itself for n times until we find the number n that gives us 1024.

So, 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2, or 2⁹, is 512, multiplied again by 2 finally gives 1024.

This means that the 10th generation in this hypothetical would have a population of 1024.

As you can see, exponents and logarithms are useful for trying to figure out a population’s growth pattern.

Problems!

If you want to actually understand these concepts, it is important to learn by doing!

Here are some simple problems you can try:

Simplify 3⁴ × 3².
Find log₁₀(1000).
Convert 3ⁿ = 27 into logarithmic form.
Solve for x: 3³ = x
Solve for x: 2ˣ = 32.
Solve for x: x² = 25

If you want, you can drop your answers in the comments!

Real World Usage

Exponents and logarithms are two core concepts that themselves offer a bunch of new insight into how the world works, but they also pave the way for your learning of more advanced mathematical concepts, like calculus and data science.

From calculating mortgage payments to measuring sound waves, these concepts pop up all around us, we just have to look for them.

Conclusion

To wrap up, I want to stress just how important exponents and logarithms are, and how many doors they open towards countless applications in math, applied sciences and beyond.

I hope that this explainer has answered some of your basic questions about these concepts, and I want to emphasize again that learning by doing, i.e. practicing, is the best and, in this case, only way to truly understand these kinds of concepts.

So, I encourage you to go and try to solve the problems I listed above, perhaps leave a comment with any questions you may have or simply share some of your own knowledge with others.