iGCSE Further Pure Mathematics Edexcel
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54 topics in 10 modules
☑️ Core 10 topics
- Logarithmic functions
- The quadratic function
- Identities and inequalities
- Graphs
- Series
- The binomial series
- Scalar and vector quantities
- Rectangular Cartesian coordinates
- Calculus
- Trigonometry
☑️ Logarithmic Functions and Indices 4 topics
- Functions a^x and log_b x
- Use and Properties of Indices and Logarithms
- Simple Manipulation of Surds
- Rationalising the Denominator
☑️ The Quadratic Function 3 topics
- Manipulation of Quadratic Expressions
- Roots of a Quadratic Equation
- Simple Examples Involving Functions of the Roots of a Quadratic Equation
☑️ Identities and Inequalities 5 topics
- Simple Algebraic Division
- The Factor and Remainder Theorems
- Solutions of Equations
- Simple Inequalities (Linear and Quadratic)
- Graphical Representation of Linear Inequalities in 2 Variables
☑️ Graphs 2 topics
- Graphs of Polynomials and Rational Functions with Linear Denominators
- Solution of Equations and Transcendental Functions by Graphical Methods
☑️ Series 3 topics
- Use of Summation Notation
- Arithmetic and Geometric Series
- Use of the Binomial Series (1 + x)^n
☑️ Scalar and Vector Quantities 7 topics
- Addition and Subtraction of Coplanar Vectors
- Multiplication of a Vector by a Scalar
- Components and Resolved Parts of a Vector
- Magnitude of a Vector
- Position Vector
- Unit Vector
- Use of Vectors to Establish Simple Properties of Geometrical Figures
☑️ Rectangular Cartesian Coordinates 5 topics
- The Distance between Two Points
- The Point Dividing a Line in a Given Ratio
- Gradient of a Straight Line Joining Two Points
- The Straight Line and its Equation
- The Condition for Two Lines to be Parallel or Perpendicular
☑️ Calculus 7 topics
- Differentiation and Integration of Sums of Multiples of Powers of x
- Differentiation of a Product, Quotient and Simple Cases of a Function of a Function
- Applications to Simple Linear Kinematics and to Determination of Areas and Volumes
- Stationary Points and Turning Points
- Maxima and Minima
- Equations of Tangents and Normals to the Curve y=f(x)
- Application of Calculus to Rates of Change and Connected Rates of Change
☑️ Trigonometry 8 topics
- Radian Measure
- Three Basic Trigonometric Ratios of Angles of any Magnitude and their Graphs
- Applications to Simple Problems in Two or Three Dimensions
- Use of Sine and Cosine Formulae
- The Identity cos^2 θ + sin^2 θ = 1
- Use of Identity tan θ = sin θ / cos θ
- Use of Basic Addition Formulae of Trigonometry
- Solution of Simple Trigonometric Equations for a Given Interval
iGCSE Further Pure Mathematics Edexcel Revision Content
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iGCSE Further Pure Mathematics Edexcel - Core - Logarithmic functions Content Preview
Core
Logarithmic functions
Logarithmic functions
Basic Definition and Properties
- A logarithm is the power to which a certain number, called the base, must be raised to obtain a given number.
- The expression is written as log_b(a) = n where b is the base, a is the number and n is the power.
- Logarithmic functions are the inverses of exponential functions.
Basic Rules of Logarithms
- Product rule: The log of a product is the sum of the logs of its factors, i.e., log_b(a * c) = log_b(a) + log_b(c).
- Quotient rule: The log of a quotient is the difference between the logs of the numerator and the denominator, i.e., log_b(a / c) = log_b(a) - log_b(c).
- Power rule: The log of an exponent is the exponent times the log of the base, i.e., log_b(a^n) = n * log_b(a).
Change of Base Formula
- Any logarithm can be computed using any other base through the change of base formula, as long as both the bases are positive and not equal to 1.
- The change of base formula is written as log_b(a) = log_c(a) / log_c(b) where a, b, c are positive numbers and b ≠ 1.
Natural Logarithm
- The natural logarithm or ln is a logarithm in the base e, where e is an irrational and transcendental number approximately equal to 2.71828.
- Natural logarithms have similar properties to those mentioned above for basic logarithms.
Solving Logarithmic Equations
- To solve equations involving logarithms, use the rules of logarithms to simplify the equations.
- If the logs in the equation have the same base, you can set the expressions inside the logs equal to each other and solve the resulting equation.
- In some cases, it may be helpful to rewrite the logarithmic equation as an exponential equation.
Relationship with Exponential Functions
- The set of all exponential functions is the inverse of the set of all logarithmic functions, and vice versa.
- The graph of a logarithm function is a reflection of the graph of the corresponding exponential function over the line y = x.
Question: If log_base 4 (8) = x, what is the value of x when expressed in terms of log_base 2?
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