Exponentials and logarithms · Year 13
In Year 13 you learn: laws of logarithms and solving exponential equations.
Exponentials like 2ˣ and eˣ model growth and decay: populations, savings, radioactivity, cooling. Logarithms are their inverse — log₂ 8 = 3 because 2³ = 8 — and they let you solve equations where the unknown is in the power.
Key facts
- aˣ = b means the same as logₐ b = x.
- log a + log b = log(ab), log a − log b = log(a/b), n log a = log(aⁿ).
- logₐ a = 1 and logₐ 1 = 0.
- ln x is the logarithm to base e (about 2.718): ln(eˣ) = x and e^(ln x) = x.
- Solve aˣ = b by taking logs of both sides: x = (log b)/(log a).
- In y = Ae^(kt), k > 0 means growth and k < 0 means decay.
Worked examples
Example 1
Solve 5ˣ = 40, giving x to 3 significant figures.
Answer: x = 2.29
Example 2
Solve log₂ x + log₂(x − 2) = 3.
Answer: x = 4
Example 3
A sample decays so that its mass is m = 80e^(−0.05t) grams after t years. When will 20 g be left?
Answer: After about 27.7 years