IB Maths AI SL Topic 1 — Number & Algebra Paper 1 & 2 In formula booklet ~8 min read

Introduction to Logarithms

A logarithm is just a way of asking a power question backwards. Instead of “what’s 2 to the power 5?” (answer: 32), a log asks “what power of 2 gives me 32?” (answer: 5). That’s the whole idea. Once you can flip between an exponential statement (like 2⁵ = 32) and a logarithmic one (log₂(32) = 5), the rest is just using your GDC.

📘 What you need to know

The flip — the most important fact: a^x = b ⟺ x = log_a(b). These two statements say exactly the same thing, just rearranged.
Read a log as a question: log_a(b) means “what power of a gives b?” The answer is the x in a^x = b.
The base rules: in log_a(b), the base a > 0 and a ≠ 1; the argument b must be > 0. You cannot take the log of zero or a negative number.
Common log: when no base is written, the base is 10. So log(x) means log₁₀(x).
Natural log: ln(x) means log_e(x), where e ≈ 2.718 (Euler’s number). Comes up everywhere in growth and decay.
Logs and powers undo each other: log_a(a^x) = x, and a^log_a(x) = x. Useful shortcut when one operation cancels the other.
The flip is in the formula booklet: you’ll find “a^x = b ⟺ x = log_a(b)” listed there — but you should know it instantly, not look it up.
Your GDC has three buttons: log (base 10), ln (base e), and log_a(b) (any base — you type in the base yourself).

The big idea — a log is just a power question, rearranged

The flip — formula booklet a^x = b ⟺ x = log_a(b)

The base, the power, and the answer all appear in BOTH forms — just in different places. Always ask “what power of the base gives the argument?” when you read a log.

So if a question asks for log₃(81), read it as “what power of 3 gives 81?” Try a few: 3¹ = 3, 3² = 9, 3³ = 27, 3⁴ = 81. So log₃(81) = 4. No calculator needed when the answer is a whole number.

Two special logs you’ll see all the time

Common log — base 10

log(x) = log₁₀(x)

When no base is shown, it’s base 10. e.g. log(100) = 2 because 10² = 100. Linked to standard form — log(a × 10ⁿ) gives roughly n for the exponent.

Natural log — base e

ln(x) = log_e(x)

e ≈ 2.718 is Euler’s number — the “natural” growth rate. Used in continuous compound interest, population growth, radioactive decay. Press the ln button on your GDC.

GDC reminder: most calculators have three log-related keys — log (base 10), ln (base e), and a separate log_a(b) entry for typing in any base you like. If your calculator only has log and ln, you can still find any other log using the change-of-base formula (covered in the next note).

🧭 Recipe — solving any log or exponential problem

Identify which form you have: is it written as a^x = b (exponential), or log_a(b) = x (logarithmic)?
Decide what’s unknown: which letter are you trying to find — the power, the answer, or the base?
If the power is unknown: flip to log form. a^x = b becomes x = log_a(b), then evaluate on your GDC.
If the answer is unknown: flip to exponential form. x = log_a(b) becomes b = a^x, then evaluate.
Try inspection first: before reaching for the GDC, check if the answer is a small whole number (e.g., log₂(64) = 6 by spotting 2⁶ = 64). Saves time and shows method clearly in working.

Worked examples

WE 1

Evaluate by inspection (whole-number answer)

Find the value of log₂(32) without using a calculator.

Step 1 — read it as a question log_2(32) asks: “what power of 2 gives 32?” Step 2 — work through powers of 2 until you hit 32 2¹ = 2 2² = 4 2³ = 8 2⁴ = 16 2⁵ = 32 ✓ log₂(32) = 5 whenever the argument is a power of the base, the answer is a whole number — no calculator needed. Build your “powers of 2” reflex: 2, 4, 8, 16, 32, 64, 128, 256.

WE 2

Converting between forms

Write the equation 4³ = 64 in logarithmic form.

Step 1 — identify the base, power, and answer 4³ = 64 base = 4, power = 3, answer = 64 Step 2 — flip using a^x = b ⟺ x = log_a(b) a = 4 (the base stays as the base) b = 64 (the answer goes inside the log) x = 3 (the power becomes the log value) log₄(64) = 3 in exam questions that ask you to “rewrite” between forms, just identify the three numbers and slot them in. The BASE always stays in the same role. The trickiest part is remembering that the answer goes INSIDE the log brackets.

WE 3

Common log — no base shown

Find log(1000) without a calculator.

Step 1 — when no base is shown, the base is 10 log(1000) means log_10(1000) Step 2 — read as a question “what power of 10 gives 1000?” 10¹ = 10, 10² = 100, 10³ = 1000 ✓ log(1000) = 3 handy shortcut: log of a power of 10 is just the exponent. log(10) = 1, log(100) = 2, log(1000) = 3, log(1 000 000) = 6. log(0.1) = −1, log(0.01) = −2.

WE 4

Logs and powers undo each other

Evaluate ln(e⁴).

Step 1 — recognise the shortcut ln means log_e (natural log, base e) log_a(a^x) = x — the log undoes the power Step 2 — apply directly ln(e⁴) = log_e(e⁴) = 4 ln(e⁴) = 4 no GDC needed. Whenever you see a log whose argument is the base raised to some power, the answer is just that power. e.g. log(10⁷) = 7, log_5(5²⁵) = 25, ln(e⁻³) = −3.

WE 5

Solve for the unknown power (GDC needed)

Solve 5^x = 80, giving your answer to 3 significant figures.

Step 1 — the power is unknown, so flip to log form 5^x = 80 ⟺ x = log_5(80) Step 2 — try inspection first 5² = 25 (too small) 5³ = 125 (too big) so x is between 2 and 3 — not a whole number, use GDC Step 3 — type log_5(80) on your GDC log_5(80) = 2.72270… x ≈ 2.72 (3 s.f.) always sanity-check with inspection first: since the answer is between 5² and 5³, x must be between 2 and 3 — and 2.72 fits. If your GDC gave you 0.27 or 5.4 you’d know to recheck.

WE 6

Real-world — bacteria doubling time

A bacterial culture doubles every hour. Starting with 1 bacterium, find how many hours until the population reaches 1 000 000. Give your answer to 3 significant figures.

Step 1 — set up the equation after t hours, population = 2^t we want 2^t = 1 000 000 Step 2 — flip to log form t = log_2(1 000 000) Step 3 — sanity-check with inspection 2¹⁰ = 1024 ≈ 10³ 2²⁰ ≈ 10⁶ — so t will be near 20 Step 4 — evaluate on GDC log_2(1 000 000) = 19.9315… t ≈ 19.9 hours the famous “2¹⁰ ≈ 10³” rule of thumb means every 10 doublings ≈ ×1000. So going from 1 to a million (×10⁶) takes about 20 doublings — matches our answer.

💡 Top tips

Always read a log as a question: “log_a(b) = what power of a gives b?” — drilling this phrase out loud trains your brain to flip automatically.
Try inspection before GDC: if the argument is a power of the base (like log₂(64), log₃(81), log(10000)), the answer is a whole number you can find by trial. Faster than typing.
Sanity-check with bounds: if 5² = 25 and 5³ = 125, then log₅(80) must be between 2 and 3. Useful for catching GDC input errors.
Remember the “undo” shortcuts: log_a(a^x) = x and a^log_a(x) = x. They cancel each other like + and − or × and ÷.
Watch the base notation: log (no base) = base 10; ln = base e; log₂, log₃, etc. — base shown explicitly. Mixing these up is the #1 source of wrong answers.

⚠ Common mistakes

Forgetting “log” means base 10: writing log(7) and using base e on your GDC will give the wrong answer. log with no base = base 10. If you want natural log, write ln.
Putting the base and argument in the wrong places when flipping: 4³ = 64 becomes log₄(64) = 3, NOT log₆₄(4) = 3 or log₄(3) = 64. The base stays the base; the answer goes inside the log.
Taking the log of zero or a negative number: log(0), log(−5), ln(−2) are all UNDEFINED. The argument must be positive. Some GDCs give an error; others give a complex number — neither is valid in AI SL.
Using a base of 1 or a negative base: log₁(anything) is undefined (since 1 to any power is just 1). The base must be positive AND not equal to 1.
Assuming logs distribute over + and −: log(a + b) is NOT log(a) + log(b). The rules for combining logs (covered next) only work for products, quotients, and powers — NEVER for sums.

Up next: Solving Exponential Equations. Once you can flip between forms, you can solve any equation where the unknown is in the power — like working out how long an investment takes to double, or when a population reaches a target. Same idea, more interesting problems.

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