Exponentiation answers the question: what happens when you multiply a number by itself a certain number of times? For example, $2^3$ is asking you to multiply 2 by itself 3 times to get 8.

A logarithm flips this question around. It asks: to get 8, how many times do we need to multiply 2 by itself? The answer, of course, is 3. We write this as:

A logarithm, or "log," is simply an exponent. It's the exponent you need to raise a specific base to in order to get a certain number. The two equations, $2^3 = 8$ and $\log_2(8) = 3$, describe the exact same relationship.

The general form looks like this: If $b^y = x$, then $\log_b(x) = y$. Here, 'b' is the base, 'y' is the exponent, and 'x' is the result.

While you can use any positive number as a base for a logarithm, two bases are so common they have their own special notation.

Common Logarithm: This is a logarithm with base 10. It's used so often in science and engineering that if you see a log written without a base, like $\log(100)$, you can assume the base is 10. Your calculator's "LOG" button uses base 10.

$\log(1000)$ is asking, "10 to what power gives 1000?" Since $10^3 = 1000$, the answer is 3.

Natural Logarithm: This is a logarithm with base $e$, a special irrational number approximately equal to 2.718. The number $e$ shows up naturally in processes involving continuous growth, like compound interest or population dynamics. The natural log is written as $\ln$. So, $\log_e(x)$ is the same as $\ln(x)$. Your calculator has a dedicated "LN" button for this.

$\ln(e)$ asks, "$e$ to what power equals $e$?" The answer is simply 1.

Before electronic calculators, performing calculations with very large or very small numbers was incredibly difficult and time-consuming. Imagine multiplying two 10-digit numbers by hand.

In the early 17th century, Scottish mathematician John Napier introduced logarithms as a tool to simplify this work. He discovered a way to turn tedious multiplication problems into much simpler addition problems, and division into subtraction. This breakthrough revolutionized fields like astronomy, navigation, and engineering, allowing for calculations that were previously impractical.

Today, even with calculators, logarithms are essential. They are used to measure the intensity of earthquakes (Richter scale), the acidity of solutions (pH scale), and the loudness of sound (decibels). These scales help us make sense of quantities that vary over an enormous range.

Let's check your understanding of these foundational concepts.

Understanding what a logarithm is—an exponent—is the most important first step. With this foundation, you'll be ready to explore what you can do with them.