Exponents and logarithms in real-world models (IB Math AI SL)
Updated October 2026
Quick answer: In SL AI, logarithms are the tool for finding an unknown power, usually time in a growth or decay model such as N = a × bᵗ. Rearrange to bᵗ = N ÷ a, then t = log(N ÷ a) ÷ log b. Or graph the model on the GDC and find where it meets the target value.
SL AI rarely tests logs abstractly — it tests them in exponential growth/decay: bacterial cultures, radioactive decay, depreciation. Know how to solve a·bx = c using logs.
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N = a × bᵗ: a is the starting value and b the growth factor per time unit. b = 1.08 means 8% growth; b = 0.92 means 8% decay.
log₁₀ x = y means 10ʸ = x, and ln x = y means eʸ = x. Both are on the calculator.
To solve bᵗ = c, take logs of both sides: t log b = log c, so t = log c ÷ log b. Any base works, if you use the same one on both sides.
Read the question for whole numbers: "after how many complete hours" means round up to the next whole hour, and check the value either side.
A logarithmic scale (pH, decibels, the Richter scale) adds 1 for every multiplication by 10.
Worked example
A culture has 500 bacteria, and the number grows by 8% every hour. Write down a model for the number N after t hours, and find when the number first reaches 2000.
Growth by 8% means multiplying by 1.08 each hour, so N = 500 × 1.08ᵗ.
Set N = 2000: 500 × 1.08ᵗ = 2000, so 1.08ᵗ = 4.
Take logs: t ln 1.08 = ln 4, so t = ln 4 ÷ ln 1.08 ≈ 18.0 hours.
Check: after 18 hours N ≈ 1998 and after 19 hours N ≈ 2158, so the number reaches 2000 just after 18 hours.
Answer: N = 500 × 1.08ᵗ; it reaches 2000 after about 18.0 hours (during the 19th hour).
How do you solve an exponential equation using logs?
Get the power on its own, as bᵗ = c. Then take the logarithm of both sides and use log(bᵗ) = t log b, so t = log c ÷ log b. Check the answer by substituting it back in.
Where can I practise exponents and logarithms in real-world models questions?
On the Logarithms and Exponentials questions page: 16 exam-style SL AI questions with full mark schemes, plus free step-by-step worked examples.
Can I use my calculator for exponents and logarithms in real-world models?
Yes: a GDC is allowed on every SL AI paper, so practise the calculator method as well as the working.