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Mathematical Methods
Exponentials and logarithms
Chapter 11
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RESTS ON
Highest common factor
CH 2
Simplify a fraction
CH 2
Form a function's maximal real domain by excluding zeros of denominators, requiring radicands to be non-negative, requiring denominator radicands to be positive, and intersecting all component restrictions
CH 6
Solve two-step linear equations by inverse operations in reverse order
CH 1
Solve equations with a scaled bracket by expanding or dividing first
CH 1
Signed numbers
CH 2
Substitute known values into a formula and evaluate it correctly
CH 1
Classify a relation as one-to-one, many-to-one, one-to-many or many-to-many from repeated inputs and outputs or from horizontal- and vertical-line evidence
CH 6
Decide whether the inverse of a function is also a function by testing whether the original is one-to-one, and choose a valid monotone domain restriction when needed
CH 6
Identify endpoints or central points, domain, range, orientation and correspondence for square-root and cube-root functions and their simple transformations
CH 6
Solve equations with the unknown inside a fraction by clearing the denominator first
CH 1
Evaluate log₁₀ of an arbitrary positive number with technology
FOUNDATIONS
Rounding
CH 3
Determine the exact coefficient carriers of a simple function or relation from enough point, feature or image information, and verify the resulting rule at every supplied condition
CH 6
Solve a pair of simultaneous linear equations for the shared (x, y)
CH 1
Build a table of values from a formula by independent substitution
CH 1
Decide whether a given transformation linearises a data set
FOUNDATIONS
Connect roots of an equation, zeros of its polynomial, linear factors and horizontal-axis intercepts, including a repeated root
CH 3
Map exact points and endpoint inclusion under translations, reflections in either axis and dilations from either axis
CH 6
Interpret f, f(x), image, preimage, rule, domain and co-domain; evaluate a function at a stated exact input and solve a simple equation of the form f(x)=c
CH 6
Apply a percentage increase or decrease
FOUNDATIONS
Percent as parts per hundred
FOUNDATIONS
Translate a described situation into a linear equation in one unknown
CH 1
Index laws and rational indices
Logarithm laws and inverse identities
Exponential equations using logarithms
Exponential and logarithmic inverses
Exponential and logarithmic models
Simplify products, quotients and powers with a common positive base by adding, subtracting or multiplying exponents as appropriate, without applying an index law across addition or subtraction
Convert exactly between negative or rational powers and reciprocal radical forms, and evaluate constructions whose required roots are exact
Multiply and divide fractions
Extract maximal perfect-power factors from a radicand and place the result in canonical coefficient-times-squarefree-radical form
Combine or expand same-base logarithms using the product, quotient and power laws, including the exact values of the logarithm of one and of its base
Apply log_a(a^x)=x and a^(log_a(x))=x under their domain conditions, including after an exact index-law rearrangement
Enforce a positive logarithm argument and a positive base unequal to one when evaluating, simplifying or solving
Translate between a^x=n and log_a(n)=x, keeping base, exponent and positive argument in their correct roles
Rewrite both sides of an exponential equation with one admissible base, equate the exponents and solve the resulting linear equation exactly
Solve a same-base exponential inequality, preserving the order for a base above one and reversing it for a base strictly between zero and one
Reduce an equation involving two powers of the same exponential quantity to a polynomial in a positive substitution, solve it, reject non-positive substitution values and back-substitute
Write a power of 10 and state its logarithm, and reverse the process
Solve equations with unknowns on both sides by collecting them onto one side
Solve a one-variable linear inequation, reversing the order relation exactly when multiplication or division by a negative value occurs
Identify a repeated subexpression, replace it with a temporary variable, solve the resulting quadratic and substitute back
Use the inverse relationship between a positive-base exponential and its same-base logarithm to swap domain and range, swap point coordinates and exchange horizontal and vertical asymptotes by reflection in y=x
Identify growth or decay, horizontal asymptote, intercepts, domain, range, one-to-one status and end behaviour of a basic or transformed exponential graph
Interchange input and output variables, rearrange for the new output, choose the branch licensed by the original restriction, and state the exchanged domain and range
Construct the inverse of a relation by interchanging every ordered pair, hence interchanging domain and range, and map graph points through reflection in y=x
Express an arbitrary-base logarithm as a quotient of two logarithms in one common calculator base and evaluate it to a stated decimal-place accuracy
Solve an exponential equation that cannot be reduced to a common base by taking logarithms consistently, isolate the unknown and distinguish the exact logarithmic form from a guarded decimal approximation
Combine logarithms where needed, convert to exponential form, solve the resulting algebraic equation and reject every candidate outside the original logarithm domains
Determine the number and approximate location of solutions of an exponential or logarithmic equation over a stated interval from a graph or an exact sign-changing enclosure, and verify the reported interval
Identify orientation, vertical asymptote, intercepts, domain, range, one-to-one status and end behaviour of a basic or transformed logarithmic graph
Connect the parameters in a normalised exponential or logarithmic rule with translations, axis reflections and horizontal or vertical dilations, and map asymptotes and exact landmark points accordingly
Determine exact parameters of a member of a stated exponential family from sufficient point, asymptote or multiplier information and verify every supplied condition
Interpret the initial value, per-period multiplier, growth or decay rate, horizontal asymptote or long-run value, and the time unit in a model of the form L+A b^t
Evaluate a stated exponential model at an admissible time and interpret the predicted amount or change without rounding an intermediate power
Solve for a time, doubling time or half-life from an exponential model, respecting the model domain, the stated time unit and exact-versus-approximate answer requirements
Distinguish a multiplicative constant-ratio pattern from additive, power-law and neither patterns in tables, graphs or described data, with attention to a stated long-run offset
Rearrange to zero, factorise, apply the null factor law and solve the resulting linear equations
Obtain the domain and range of a relation represented by ordered pairs, a table, a rule or described graph data, preserving set membership and open or closed endpoints
Read a transformed rule in normalised form and name its horizontal and vertical translations, axis reflections and reciprocal horizontal or direct vertical dilation factors
Apply or undo a sequence of transformations in a stated order, with dilations and reflections acting before translations in the normalised parameter form
Percentage change as a multiplier
Turn an equation's exact solution into the answer the situation demands