Logarithms, Exponents and Equations
Apply exponent and logarithm laws exactly and convert between equivalent forms to isolate an unknown.
How to study Higher Mathematics
Practise a skill without a calculator first, connect it to neighbouring techniques, then apply it in unfamiliar contexts with complete working and interpretation.
Core concepts
Concept 1
index laws
Exam cue: Can terms be written with a common base?
Concept 2
log laws
Exam cue: Which log law combines the expression?
Concept 3
change between forms
Exam cue: Are all resulting log arguments positive?
Concept 4
logarithmic equations
Concept 5
exponential equations
Risk pitfalls and guardrails
Distributing a logarithm over addition
Guardrail: Check signs, brackets, domain restrictions, units and whether the answer needs justification.
Dropping a power when moving it in front of a log
Guardrail: Check signs, brackets, domain restrictions, units and whether the answer needs justification.
Accepting a solution outside the logarithm domain
Guardrail: Check signs, brackets, domain restrictions, units and whether the answer needs justification.
Memory anchors
Product becomes sum
log(ab) = log a + log b.
Quotient becomes difference
log(a/b) = log a − log b.
Power comes forward
log(aᵏ) = k log a.
Common base, compare powers
Rewrite exponential terms before equating exponents.
Log undoes exponent
Convert aˣ = b to x = logₐb when appropriate.
Arguments stay positive
Check every logarithm input after solving.
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
Evaluate log₂32.
Evaluate log₃(1/27).
Answer all questions to submit.
Next step personalized recommendations
Continue learning
Move forward only after this module is stable.
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