Indices, Logarithms and Matrices
Indices provide a compact way to write repeated multiplication, logarithms reverse exponentiation, and matrices organise numerical information in rows and columns.
Learning objectives
After studying these notes, learners should be able to define the main terms, apply the central rules and formulas, solve structured examples and explain each answer clearly.
Key ideas
- Positive, zero, negative and fractional indices
- Laws of logarithms and change between index and logarithmic forms
- Order and equality of matrices
- Addition, subtraction and scalar multiplication
- Multiplication and determinant of a two-by-two matrix
Essential rules and formulas
- a^m x a^n = a^(m+n)
- a^m / a^n = a^(m-n)
- log(xy) = log x + log y
- For [[a,b],[c,d]], determinant = ad – bc
Worked example
Simplify 2^3 x 2^-5 = 2^-2 = 1/4. For A = [[2,1],[3,4]], det(A) = 8 – 3 = 5.
Study method
Begin every problem by listing the information supplied and identifying the rule that connects it to the unknown quantity. Keep algebraic steps on separate lines, include units where required and check that the final result is reasonable.
Practice questions
- Simplify 27^(2/3).
- Solve log base 2 of x = 5.
- Multiply [[1,2],[0,3]] by [[2,1],[4,0]].
Revision checklist
- Can you state each definition without looking at the notes?
- Can you select the correct formula and substitute values accurately?
- Can you explain the reasoning behind each step?
- Can you solve a similar examination question independently?
