IB Maths AI HLExponentials & LogsPaper 1 & 2logab~6 min read
Introduction to Logarithms
A logarithm is the inverse of a power — it answers the question “what power do I raise this base to?” That makes logarithms the tool for solving equations where the unknown sits in the exponent.
📘 What you need to know
A logarithm is the inverse of a power: it finds the exponent.
ax = b is the same statement as x = logab, for a > 0, b > 0, a ≠ 1.
a is the base; logab is read “log base a of b”.
The natural logarithm ln x has base e ≈ 2.718; log x written with no base means base 10.
Equations with the unknown in the power are exponential equations.
Solve a neat one by inspection; otherwise rewrite it as a logarithm and use your GDC.
What is a logarithm?
A power such as ax = b links three numbers: the base a, the exponent x and the result b. A logarithm rearranges that statement to make the exponent the subject.
The power statement 23 = 8 and the logarithm log28 = 3 say exactly the same thing — a logarithm isolates the exponent.
Logarithm — the inverse of a powerax = b ⇔ x = logabfor a > 0, b > 0, a ≠ 1 — this relationship is in the formula booklet
Natural logs and base 10
Two bases appear so often they get their own notation. The natural logarithm uses the constant e; the common logarithm, written with no base shown, uses base 10.
Two special logarithms
ln x = logex · log x = log10xe ≈ 2.718 (Euler’s number) — your GDC has ln, log and a logab key
Solving exponential equations
An exponential equation has the unknown in the power. If the result is a neat power of the base, solve it by inspection — 2x = 8 clearly gives x = 3.
When the result is not a neat power — like 2x = 10 — rewrite the equation as a logarithm, x = log210, and evaluate it on the GDC.
GDC tip: for any base, use the logab key — enter the base and the number directly. The ln and log keys handle base e and base 10.
🧠Recipe — evaluating a log or solving a x = b
Identify the base — the number being raised to a power.
Ask “what power of the base gives the result?” — try inspection first.
If it is a neat power of the base, state the answer directly.
If not, rewrite ax = b as x = logab.
Evaluate logab on the GDC, rounding as the question asks.
Worked examples
WE 1
Evaluating a logarithm by inspection
Find the value of log381.
ask: what power of 3 gives 81?31=3, 32=9, 33=27, 34=8134 = 81, so the power is 4log381 = 4a logarithm is just “the power” — here, the power of 3 that makes 81.
WE 2
Switching between the two forms
(a) Write 54 = 625 in logarithmic form. (b) Write log7343 = 3 in exponential form.
use ax = b ⇔ x = logab(a) base 5, power 4, result 62554 = 625 ⇒ log5625 = 4(b) base 7, log equals 3, result 343log7343 = 3 ⇒ 73 = 343(a) log5625 = 4 · (b) 73 = 343the base stays the base; the log value is the power.
WE 3
An exponential equation by inspection
Solve the equation 3x = 243.
is 243 a neat power of 3?34 = 81, 35 = 243the power that works is 5x = 5when the result is a clean power of the base, no logarithm is needed.
WE 4
An exponential equation needing a log
Solve the equation 5x = 90, giving your answer correct to 3 significant figures.
90 is not a neat power of 5 — use a log5x = 90 ⇒ x = log590evaluate on the GDCx = 2.79588…x ≈ 2.80 (3 s.f.)rewrite ax = b as x = logab, then use the logab key.
WE 5
Natural log and base 10
Solve, correct to 3 significant figures: (a) ex = 20 and (b) 10x = 0.05.
(a) base e — the log is lnex = 20 ⇒ x = ln 20 = 2.99573…(b) base 10 — the log is log10x = 0.05 ⇒ x = log 0.05 = −1.30103…(a) x ≈ 3.00 · (b) x ≈ −1.30the log of a number less than 1 is negative — that is fine.
WE 6
Full question: bacteria growth
A colony of bacteria doubles every hour. Starting from 50 bacteria, the number after t hours is N = 50 × 2t. (a) Find N after 4 hours. (b) Find the exact time when N = 3200. (c) Find the time when N = 5000, to 3 s.f.
(a) substitute t = 4N = 50 × 24 = 50 × 16 = 800(b) 50 × 2t = 3200 ⇒ 2t = 6426 = 64, so t = 6 (by inspection)(c) 50 × 2t = 5000 ⇒ 2t = 100t = log2100 = 6.6439…(a) 800 · (b) t = 6 hours · (c) t ≈ 6.64 hoursdivide out the 50 first, then inspect or take a log of 2t = …
💡 Top tips
Read logab as a question: “what power of a gives b?”
Try inspection first — if the result is a neat power of the base, you need no log.
Rewrite ax = b as x = logab — this relationship is in the formula booklet.
Use ln for base e and log for base 10; the logab key handles any other base.
A logarithm can be negative — it happens whenever the number is between 0 and 1.
âš Common mistakes
Confusing the base and the number — log381 finds a power of 3, not of 81.
Trying to take a log of zero or a negative number — only b > 0 is allowed.
Forgetting “log” with no base means base 10, not base e.
Rounding too early — keep the GDC value, then round the final answer.
Thinking a negative log is an error — log 0.05 = −1.30 is perfectly valid.
Next up: Laws of Logarithms — rules for adding, subtracting and scaling logs, which mirror the index laws exactly. They let you combine several logarithms into one and solve far richer equations.
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