Exponentials and Logarithms
Interpret exponential graphs, apply logarithm laws and solve equations reducible to exponential form.
How to prepare for the current ESAT
Confirm your course combination, build no-calculator fluency from Mathematics 1 upward, and use official Pearson materials for authentic timing and interface rehearsal.
Core concepts
Concept 1
Connect aˣ = b with x = logₐb and the exponential graph.
Exam cue: Check the base and positivity conditions.
Concept 2
Combine or expand logarithms using product, quotient and power laws.
Exam cue: Factor or substitute a repeated exponential expression.
Concept 3
Use algebraic substitution to reduce compound exponential equations.
Exam cue: Verify the solution in the original equation.
Risk pitfalls and guardrails
Using log(x + y) = log x + log y.
Guardrail: Do not use calculator-dependent shortcuts, import a fact outside the annual specification or overlook a unit, sign, scale or course-module boundary.
Ignoring that logarithm arguments must be positive.
Guardrail: Do not use calculator-dependent shortcuts, import a fact outside the annual specification or overlook a unit, sign, scale or course-module boundary.
Applying an unavailable change-of-base requirement.
Guardrail: Do not use calculator-dependent shortcuts, import a fact outside the annual specification or overlook a unit, sign, scale or course-module boundary.
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
A real logarithm requires a positive argument.
Repeated power
Use a substitution such as u = aˣ for a quadratic form.
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
log₂32 equals?
3ˣ=81 gives x=?
Answer all questions to submit.
Next step personalized recommendations
Continue learning
Move forward only after this module is stable.
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