Introduction to Logarithms

Hi! Today we’re learning about logarithms (or just “logs” for short). At first they look weird and a bit scary — but here’s the secret: a logarithm is just the opposite of a power. It’s the same idea as how subtraction is the opposite of addition, or division is the opposite of multiplication. Once you see this, the whole topic clicks.

Step 1: What is a logarithm, really?

Let’s start with a simple question. If I told you that 2 raised to some power gives you 8, what is that power?

2 raised to what = 8?

You can probably figure this out in your head: 2³ = 8, so the answer is 3. That’s exactly what a logarithm does — it asks the question:

“What power do I need to raise the base to, in order to get this number?”

And we write that question like this:

Read it as: “log base 2 of 8 equals 3.” It’s saying “the power of 2 that gives you 8 is 3.”

Step 2: Powers and logs are opposites — see them side by side

This is the most important picture in the whole lesson. Look carefully:

These two equations say exactly the same thing — they’re just rearranged. So whenever you see one, you can flip it into the other:

The double arrow ⇔ means “if and only if” — basically these two are the same fact written two ways.

Step 3: The three rules about a, b and x

For a logarithm to make sense, there are three rules you need to keep in mind:

1. The base a must be positive: a > 0
2. The base a can’t be 1: a ≠ 1   (because 1 to any power is just 1, so it’d be useless)
3. The number b inside the log must be positive: b > 0   (you can’t take the log of a negative number or zero)

Step 4: Reading and saying logs out loud

Let’s get comfortable with the words. When you see log_a(b):

• The little number at the bottom (a) is called the base
• You say it as: “log base a of b“
• It’s asking: “what power of a gives you b?”

Some practice:

• log₅(125)  →  “log base 5 of 125”  →  “what power of 5 gives 125?”  →  3 (because 5³ = 125)
• log₃(81)  →  “log base 3 of 81”  →  “what power of 3 gives 81?”  →  4 (because 3⁴ = 81)
• log₁₀(1000)  →  “log base 10 of 1000”  →  “what power of 10 gives 1000?”  →  3 (because 10³ = 1000)
• log₂(18)  →  “what power of 2 gives 18?”  →  −3 (because 2⁻³ = 18)

Step 5: Two special logs you’ll see all the time

In IB Maths (and basically all of science), two specific bases pop up so often that they get their own shorthand notation. Let me introduce you to both.

1. The natural logarithm (ln)

This one uses a very special number called e — pronounced “ee”. e is a mathematical constant, just like π (pi). Its approximate value is:

This number is so important that scientists call it Euler’s number (after the Swiss mathematician Leonhard Euler). It shows up naturally in things like compound interest, population growth, and radioactive decay.

When the base of a log is e, instead of writing log_e(x), we just write ln x. The “ln” stands for “natural log” (it comes from the Latin logarithmus naturalis).

2. The common logarithm (log with no base)

If you see “log” with no base written, it secretly means base 10. So:

For example:

• log 100 = 2  (because 10² = 100)
• log 10 000 = 4  (because 10⁴ = 10 000)
• log 1 = 0  (because 10⁰ = 1)

Step 6: Why do logarithms even exist?

Good question! Logs were invented to solve exponential equations — equations where the unknown is in the power.

Some exponential equations are easy. You can solve them just by looking at them — we call this solving by inspection:

• 2^x = 8  →  you can see that x = 3 (because 2³ = 8)
• 3^x = 81  →  you can see that x = 4 (because 3⁴ = 81)
• 5^x = 25  →  x = 2 (because 5² = 25)

But what if the answer doesn’t pop out of your head? Like this one:

2^x = 10

Hmm. 2² = 4 (too small). 2³ = 8 (too small). 2⁴ = 16 (too big). So x is somewhere between 3 and 4 — but exactly where?

This is where logarithms save the day. We use the swap trick from Step 2:

Boom — done. The whole reason logarithms exist is to get the unknown out of the power and into a place we can work with it.

Step 7: Two handy facts you’ll use a lot

Before we do worked examples, here are two quick results that come straight from what we’ve learned. They’ll save you time in exams:

Why? Because a⁰ = 1 for any base. So the question “what power of a gives 1?” is always answered by zero.

Why? Because a¹ = a. So “what power of a gives a?” is always 1.

Quick checks:

• log₇(1) = 0 ✓
• log₁₀₀(1) = 0 ✓
• log₅(5) = 1 ✓
• ln(e) = 1 ✓  (because ln means “log base e“, and log of the base is always 1)