Exponentials and Logarithms in A-Level Maths
Exponentials & Logarithms
We study Exponentials & Logarithms at A-Level Maths after having first seen exponential graphs during GCSE Maths (at the Higher level). At GCSE, we look at the graphs of powers of integers. However, at A-Level, we look at powers of a special number $e=2.7182818284…$. The reason we study this number in particular, is because there are some simple interesting properties when we differentiate. We also look at logarithms, their graphs and equations and how they are the inverses to exponential functions.
Exponential & Log graphs, Log Equations, Growth & Decay and the Differentiation of Log Functions are covered in the first year of A-Level Maths. Not a lot more is covered in the second year except to include the exponential curve in combined transformations.
Other Areas in AS-Maths
- PROOF – proof by deduction, proof by exhaustion, disproof by counterexample
- ALGEBRA & FUNCTIONS – completing the square, cubics, curve sketching, discriminant, indices, inequalities, polynomials, quadratics, simultaneous equations, surds, transformations
- COORDINATE GEOMETRY: straight lines, equation of a circle
- SEQUENCES & SERIES – binomial expansion
- TRIGONOMETRY – non-right-angled triangles, trigonometric equations, trigonometric graphs, trigonometric identities
- DIFFERENTIATION – differentiation from first principles and differentiating polynomials, increasing & decreasing functions, stationary points, tangents & normals, differentiating e to the kx
- INTEGRATION – fundamental theorem of calculus and integrating powers of x, definite integrals
- NUMERICAL METHODS – (not covered at AS Level)
- VECTORS – two-dimensional vectors, vector arithmetic, vectors in context
Other Areas in A2-Maths
- PROOF – proof by contradiction
- ALGEBRA & FUNCTIONS – modulus of a function, partial fractions, inverse and composite functions, compound transformations
- COORDINATE GEOMETRY – parametric equations
- SEQUENCES & SERIES – arithmetic series (and sigma notation), geometric series, sequences, binomial expansion
- TRIGONOMETRY – radians, arc length & area of a sector, small angle approximations, reciprocal trigonometric functions, inverse trigonometric functions, double & compound angle formulae
- DIFFERENTIATION – concavity, convexity & inflection points, derivatives of trigonometric functions, product, quotient & chain rule, parametric & implicit differentiation, differentiating exp/log functions, differential equations & rates
- INTEGRATION – further integration, integration using trigonometric identities, integration by substitution, integration by parts, integration using partial fractions, solving differential equations,
- NUMERICAL METHODS – locating roots using iteration (including Newton-Raphson), trapezium rule
- VECTORS – 3D vectors