IB Maths Paper 1 & 2 22 min read

Laws of Logarithms

Welcome back! Now that you know what a logarithm is, it’s time to learn the laws of logarithms — a small set of rules that let you simplify, combine, and split log expressions. Think of these laws as shortcuts that turn ugly equations into clean ones. Get comfortable with them, and you’ll breeze through Paper 1 questions on logs.

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What you will learn today

The 3 main log laws — multiplication, division, and powers
Why each law works (with a simple proof)
5 useful shortcuts you should memorise
How to handle expressions with ln (natural log)
How to spot when logs are undefined (and how to avoid trick questions)
4 worked examples, from easy to exam-style

Step 1: A quick refresher

Before diving into laws, let’s quickly remember what a log is. From the Introduction to Logarithms note:

a^x = b ⇔ x = log_a(b)

A log just asks “what power did I raise the base to, in order to get this number?” Now — because logs and powers are opposites, every law of indices has a matching log law. That’s the secret pattern behind everything in this lesson.

Big picture: the index laws (multiply → add powers, divide → subtract powers, etc.) are the family of these log laws. Every log rule below is just an “upside-down” version of an index rule. So if you know your indices, half of this is already in your head.

Step 2: The 3 main laws of logarithms

There are 3 main laws you need to know. They all assume the same base (you’ll see why in a moment). Let me walk you through each one.

LAW 1

The Multiplication Law (also called the Product Law)

log_a(x y) = log_a(x) + log_a(y)

If you have a log of two things multiplied together, you can split it into the sum of two separate logs.

Example: log₂(8 × 4) = log₂(8) + log₂(4) = 3 + 2 = 5

Check: 8 × 4 = 32, and log₂(32) = 5 because 2⁵ = 32. ✓ Same answer!

LAW 2

The Division Law (also called the Quotient Law)

log_a(xy) = log_a(x) − log_a(y)

If you have a log of one thing divided by another, you can split it into the difference of two logs.

Example: log₃(819) = log₃(81) − log₃(9) = 4 − 2 = 2

Check: 819 = 9, and log₃(9) = 2 because 3² = 9. ✓ Match!

LAW 3

The Power Law

log_a(x^m) = m × log_a(x)

If the number inside the log has a power on it, you can bring the power out to the front as a multiplier.

Example: log₂(8³) = 3 × log₂(8) = 3 × 3 = 9

Check: 8³ = 512, and log₂(512) = 9 because 2⁹ = 512. ✓ Same!

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Examiner Tip

All 3 of these laws are in the IB formula booklet. So you don’t need to memorise them word-for-word — but you absolutely must practise using them. The exam tests whether you can apply them quickly, not just recite them.

Important: all 3 laws only work when the bases are the SAME. You can’t combine log₂(5) and log₃(7) using these laws — the bases don’t match. (You’d need the change of base formula, which we’ll cover later.)

Step 3: Why do these laws work?

Let me prove the multiplication law in 30 seconds — once you see this, the others will feel obvious too.

Let’s say: x = a^p and y = a^q
Then: xy = a^p × a^q = a^p+q (using the index law: same base, multiply, add powers)
Take log base a of both sides: log_a(xy) = log_a(a^p+q) = p + q
But: p = log_a(x) and q = log_a(y)
So: log_a(xy) = log_a(x) + log_a(y) ✓

And just like that, the multiplication law is proved! It comes directly from the index law for multiplying powers. The other two laws come from the same idea — just using division and “power of a power” instead.

The pattern at a glance: Index laws turn into log laws by “trading places”. Multiply turns into add. Divide turns into subtract. Power on top becomes multiply in front. Easy.

Step 4: Five useful shortcuts to memorise

Beyond the 3 main laws, there are 5 little shortcuts that pop up everywhere. They’re not in the formula booklet, but they save tons of time. Let me show you each one.

SHORTCUT 1

Log of 1 is always zero

log_a(1) = 0

For any base, the log of 1 equals 0. Why? Because a⁰ = 1 — anything to the power of zero is 1. So the question “what power gives me 1?” is always answered by zero.

Examples: log₅(1) = 0, log₁₀₀(1) = 0, ln(1) = 0

SHORTCUT 2

Log of the base equals 1

log_a(a) = 1

If the number inside the log matches the base, the answer is 1. Why? Because a¹ = a. The power of 1 always gives back the base itself.

Examples: log₇(7) = 1, log₁₀(10) = 1, ln(e) = 1

SHORTCUT 3

Log of base to a power = the power

log_a(a^k) = k

If the inside of the log is the base raised to some power, you just pull out the power. This is just the Power Law (Law 3) combined with Shortcut 2.

Examples: log₃(3⁵) = 5, log₁₀(10²) = 2, ln(e^x) = x

SHORTCUT 4

Power of a log just gives the inside number back

a^log_a(x) = x

Logs and powers cancel each other out when they have the same base. Think of it like adding 5 then subtracting 5 — you end up where you started.

Examples: 2^log₂(7) = 7, 10^log(50) = 50, e^ln(x) = x

SHORTCUT 5

Log of a fraction “1 over x” flips the sign

log_a(1x) = −log_a(x)

If the inside of the log is “1 over something”, you can pull out a minus sign. Why? Use the division law: log_a(1x) = log_a(1) − log_a(x) = 0 − log_a(x) = −log_a(x).

Example: log₂(18) = −log₂(8) = −3

Pro tip: the 5 shortcuts above aren’t in the formula booklet, but they save so much time. Make a flashcard for each and review them daily for a week — they’ll become muscle memory.

Step 5: All these laws work for ln too!

Remember from the last lesson — ln is just log base e. So every law and shortcut above works exactly the same with ln. You just replace “log_a” with “ln” everywhere.

Two particularly useful examples for IB exams:

Cancelling on the inside

ln(e^x) = x

The ln “undoes” the e^x. They cancel out and leave just x.

Cancelling on the outside

e^ln(x) = x

The e^… “undoes” the ln. They cancel out and leave just x.

You’ll use these constantly when solving exponential equations involving e. Get used to spotting the pattern.

Step 6: When are logarithms undefined?

Quick reminder from the Introduction note: you can never take the log of zero or a negative number. So if the inside of a log gives you a value that is ≤ 0, the log doesn’t exist.

This becomes important when the inside is an expression with x in it. For example:

log_a(x) is defined only when x > 0
log_a(x − 5) is defined only when x > 5 (because we need x − 5 to be positive)
log_a(2x + 1) is defined only when x > −12

Watch out! When you solve a log equation, you might get an answer that makes the original log undefined. Always check your final answers by plugging them back in. If any log becomes log of zero or a negative number, throw that solution away.

Step 7: A common trap to avoid

Here’s a mistake students make all the time. Do NOT confuse these two:

✓ Correct

log_a(xy) = log_a(x) + log_a(y)

Multiplying inside the log → add separate logs.

❌ WRONG

log_a(x + y) ≠ log_a(x) + log_a(y)

Splitting log of a sum. log_a(x + y) is NOT log_a(x) + log_a(y). The laws are for products, not sums!
Not checking final answers. If you get a solution that makes the inside of any log zero or negative, you must reject it.
Mixing different bases. log₂(5) + log₃(7) cannot be combined using these laws — the bases must match.
Forgetting to apply the power law fully. 3 log(x) means log(x³), not log(3x). The 3 becomes a power, not a multiplier inside.
Confusing log(x²) with [log(x)]². The first one becomes 2 log(x) using the power law. The second one is log(x) × log(x) — totally different, and it has no special law.
Trying to split logarithm coefficients. 2 log(3) does NOT equal log(2) × log(3). The 2 only acts as a power: 2 log(3) = log(3²) = log(9).
Cancelling logs from both sides incorrectly. If log(x) = log(y), then yes, x = y. But only if both sides are single logs with the SAME base — and both arguments are positive.

Final word from your teacher: the laws of logarithms feel intimidating because of all the symbols, but they’re really just 3 main rules + 5 quick shortcuts. The key skill is learning to spot the pattern: see a product? use Law 1. See a fraction? use Law 2. See a power? use Law 3. Once you’ve done 15–20 mixed practice questions, your brain will start applying the right law automatically. You’ve got this!

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