Exponentials, Logarithms and Growth Models
Using exponential and logarithmic functions, laws, equations, linearising transformations, growth and decay with critical evaluation of models.
How to study A-level Mathematics
Define the objects and conditions, select a representation, execute a justified method, then validate and interpret the result.
Core concepts
Concept 1
A logarithm is the inverse operation to exponentiation for a positive base other than one.
Exam cue: State positivity restrictions before taking logarithms.
Concept 2
Logarithm laws convert products, quotients and powers but do not distribute across sums.
Exam cue: Use a suitable logarithmic transformation to obtain a linear relationship and interpret its gradient and intercept.
Concept 3
Exponential models have a constant proportional rate of change and must be assessed against contextual assumptions.
Exam cue: Check whether continuous growth or decay is plausible over the modelled interval.
Risk pitfalls and guardrails
Writing log(x+y) as log x plus log y.
Guardrail: Do not replace proof with examples, model conditions with unstated assumptions, or mathematical interpretation with unverified calculator output.
Accepting a logarithmic solution outside the original domain.
Guardrail: Do not replace proof with examples, model conditions with unstated assumptions, or mathematical interpretation with unverified calculator output.
Extrapolating an exponential model far beyond the evidence without discussing limitations.
Guardrail: Do not replace proof with examples, model conditions with unstated assumptions, or mathematical interpretation with unverified calculator output.
Memory anchors
Logarithm
log_a x is the exponent to which a must be raised to obtain x.
Natural Logarithm
ln x is the logarithm to base e.
Log Product Law
The logarithm of a product is the sum of the logarithms.
Linearisation
Linearisation transforms a relationship so its parameters can be estimated from a straight line.
Exponential Model
An exponential model changes at a rate proportional to its current amount.
Checkpoint rule
Do the check-up only after you can summarize each concept in one sentence and identify one dangerous pitfall from memory.
Knowledge Check (after reading)
Short check-up to confirm understanding of this module.
Check-up Questions
Find the exponent to which 2 must be raised to obtain 32.
Simplify logₐx+logₐy.
Answer all questions to submit.
Next step personalized recommendations
Continue learning
Move forward only after this module is stable.
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