Exponentials and Logarithms
Interpret exponential graphs, apply logarithm laws and solve equations reducible to exponential form.
How to prepare for the current TMUA
Secure the shared mathematical specification, practise application in unfamiliar settings, then make every Paper 2 implication and proof step explicit.
Core concepts
Concept 1
Connect an exponential statement with its inverse logarithmic statement.
Exam cue: Check the base and positivity conditions.
Concept 2
Combine and expand logarithms using product, quotient and power laws.
Exam cue: Look for a repeated expression such as aˣ.
Concept 3
Reduce compound exponential equations through factorisation or substitution.
Exam cue: Verify candidate solutions in the original equation.
Risk pitfalls and guardrails
Splitting the logarithm of a sum.
Guardrail: Do not treat examples as proof, reverse an implication, cancel a possible zero or rely on calculator-only intuition.
Allowing a non-positive logarithm argument.
Guardrail: Do not treat examples as proof, reverse an implication, cancel a possible zero or rely on calculator-only intuition.
Using the excluded change-of-base formula as assumed knowledge.
Guardrail: Do not treat examples as proof, reverse an implication, cancel a possible zero or rely on calculator-only intuition.
Memory anchors
Inverse pair
aˣ = b exactly when x = logₐb.
Product law
logₐx + logₐy = logₐ(xy).
Quotient law
logₐx − logₐy = logₐ(x/y).
Power law
k logₐx = logₐ(xᵏ).
Log domain
Every real logarithm argument must be positive.
Repeated power
Substitute u = aˣ when an exponential equation is quadratic in aˣ.
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
If 2^(x+1)=32, what is x²−1?
The equation 9^x=27^(x−1) has solution x. What is 6x?
Answer all questions to submit.
Next step personalized recommendations
Continue learning
Move forward only after this module is stable.
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