IB Maths AA HL Topic 1 — Number & Algebra Paper 1 & 2 ~7 min read

Introduction to Logarithms

A logarithm is just the inverse of a power. It answers the question: “what exponent do I need to put on the base to get this number?” The notation is new but the idea isn’t — you’ve used it ever since you noticed that 2 × 2 × 2 = 8. The “3” in that statement is the logarithm. Once you can flip between exponential form and log form fluently, almost everything else in this section follows.

📘 What you need to know

A logarithm reverses a power. a^x = b ⇔ x = log_ab.
The base a must be positive and not equal to 1. The argument b must be positive (you can’t take the log of zero or a negative).
Read log_ab as “the power you raise a to in order to get b“.
“log” with no base shown means log₁₀. “ln” means log_e — the natural log, where e ≈ 2.718.
Easy logs come from inspection. Harder ones go through your GDC.
The flip identity a^x = b ⇔ x = log_ab is in the formula booklet — but learn it anyway, it’s the engine of everything log-related.

What does a logarithm actually mean?

Take a question like “what power of 2 gives 8?” The answer is 3, because 2³ = 8. The logarithm is just a piece of notation that lets you write “the answer to that question” cleanly:

log₂(8) = 3 means 2³ = 8

That’s it. Same statement, two notations. The “log” version is the right shape when you want the power to be the subject — which is the case any time you’re solving an exponential equation.

Read more examples like a question: log₃(81) = ? asks “what power of 3 gives 81?” → 4. log₅(125) asks “what power of 5 gives 125?” → 3.

The flip — exponential ↔ log form

The single most useful skill in this topic is being able to bounce between the two forms instantly.

The flip identity a^x = b ⇔ x = log_ab ✓ in formula booklet (where a > 0, a ≠ 1, b > 0)

Same statement, two notations

Spot the pattern: the base stays the base in both forms. The result on the right of the exponential becomes the argument inside the log. The power that was an exponent moves out and becomes the value the log equals.

Two special logarithms you’ll see everywhere

Most logs in IB exams use one of two specific bases. Both have shorthand notations.

🔟

Common log — base 10

log x means log₁₀x

e.g. log(1000) = 3

🌱

Natural log — base e

ln x means log_ex

e ≈ 2.71828… (Euler’s number)

🧠 Notation that catches everyone out

If you see log with no subscript, it means base 10. If you see ln, it means base e. Anything else needs an explicit subscript: log₂, log₅, etc. Mixing these up is the #1 reason students lose marks early on this topic.

🤔 Why is e “natural”?

Because e^x is the unique exponential that has slope equal to its own value at every point — which means ln slips into derivatives and integrals more cleanly than any other base. You’ll see this in calculus. For now: just know the notation.

Using your GDC for logs

Most calculators have three buttons:

ln → natural log (base e) | log → common log (base 10) | log_a → lets you choose any base

If your calculator only has ln and log buttons (no general-base button), you can still compute any log using the change-of-base formula — but that’s covered in the next note. For now, the dedicated log_a button is the cleanest route.

When is a logarithm undefined?

You can’t take the log of zero, and you can’t take the log of a negative number. Both rules come straight from the flip identity — there’s no real exponent that sends a positive base to zero or below.

log_ax is defined only when x > 0. Similarly, log_a(x − 4) needs x > 4.

If you’re solving a log equation and you get a value of x that makes any log inside the equation undefined, throw it out. Always check your final answers against the original equation.

Worked examples

WE 1

Evaluate by inspection

Find the value of log₂(32).

Ask the question: “what power of 2 gives 32?” 2¹ = 2, 2² = 4, 2³ = 8, 2⁴ = 16, 2⁵ = 32 ✓ log₂(32) = 5

WE 2

Natural log of an exponential

Find the exact value of ln(e⁷).

Read it as a question: “what power of e gives e⁷?” obvious answer: 7 ln(e⁷) = 7 ln and e^x cancel each other — they’re inverse operations

WE 3

Solve by inspection

Solve 3^x = 81 for x.

Step 1: Use the flip — write in log form x = log₃(81) Step 2: Read it as a question — “what power of 3 gives 81?” 3⁴ = 81 ✓ x = 4

WE 4

Solve using GDC

Solve 5^x = 200 for x, giving your answer to 3 s.f.

Step 1: Flip into log form x = log₅(200) Step 2: GDC — use the log_a button, base 5, argument 200 log₅(200) = 3.29202… x = 3.29 (3 s.f.) no nice integer answer — that’s normal once GDC is needed

WE 5

Exponential equation with base e

Solve e^x = 12 for x, giving your answer to 3 s.f.

Step 1: Flip — base is e, so the log is ln x = ln(12) Step 2: GDC — ln button ln(12) = 2.4849… x = 2.48 (3 s.f.) an exact answer would just be ln(12) — leave it like that if “exact” is asked for

WE 6

Reverse — given the log, find the number

Given that log₃(x) = 4, find the value of x.

Step 1: Flip back to exponential form log₃(x) = 4 ⇔ 3⁴ = x Step 2: Evaluate 3⁴ = 81 x = 81 the flip works both ways — practice both directions until it’s automatic

💡 Top tips

Read every log as a question. “log₄(64)” = “what power of 4 gives 64?” Out loud if you have to. It removes 90% of the confusion.
The flip is the engine. If something looks tricky, write it in the other form. Often the equation suddenly makes sense.
If a question asks for an exact answer, leave it as ln(12) or log₅(200) — don’t reach for the GDC.
If it asks for 3 s.f., then GDC away and round properly at the end.
Memorise the special notation. log = base 10, ln = base e. No exceptions.
Always double-check that the argument inside the log is positive — that’s the only domain restriction.
Keep the formula booklet open during practice. The flip identity a^x = b ⇔ x = log_ab sits there waiting for you in the exam.

⚠ Common mistakes

Confusing log and ln. “log” alone means base 10. “ln” means base e. They are not the same button.
Trying to take the log of zero or a negative number. log(0) and log(−5) are undefined. There’s no exponent that makes a positive base land there.
Reading the base as the argument. log₂(8) is “log base 2 of 8”, not “log of 28” or “log 2 times 8”. The little subscript is the base.
Flipping incorrectly. 5^x = 200 becomes x = log₅(200) — the base on the right is 5, the argument is 200. Don’t swap them.
Forgetting that ln(e) = 1. The natural log of e is 1, not e. (Because e¹ = e.)
Decimalising when “exact” is asked for. If the answer is ln(12), leave it as ln(12). 2.4849… is not exact.
Keeping invalid solutions. If solving an equation gives x = −2 but the original has log(x + 1), then x = −2 makes log(−1) — invalid. Reject it.

Once the flip becomes automatic — exponential to log, log to exponential, both directions, no thinking — the rest of this section (laws, equations, applications) just builds on top. Spend time here; it pays off everywhere later.

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