IB Mathematics AA SL Deductive Proof Numerical and Algebraic Study Notes

IB Mathematics AA SL Exponents and logarithms Study Notes

IB Mathematics AA SL Exponents and logarithms Study Notes Offer a clear explanation of Exponents and logarithms , including various formula, rules, exam style questions as example to explain the topics. Worked Out  examples and common problem types provided here will be sufficient to cover for topic Exponents and logarithms

Simple Deductive Proofs

Simple Deductive Proofs: LHS to RHS Format

A deductive proof uses known facts, properties, or identities to show that two expressions are equal. One common method is to start with the left-hand side (LHS) of an identity and simplify or manipulate it step-by-step until it becomes the right-hand side (RHS).

Key Steps in LHS to RHS Proofs:

  1. Start by writing the LHS of the identity.
  2. Apply algebraic identities or known formulas step-by-step.
  3. Justify each step (briefly if necessary).
  4. Simplify until the expression becomes identical to the RHS.
  5. Conclude by stating: LHS = RHS, hence proven.

Common Identities Used in Proofs:

  • \( a^2 – b^2 = (a – b)(a + b) \)
  • \( (a + b)^2 = a^2 + 2ab + b^2 \)
  • \( \frac{a^m}{a^n} = a^{m-n} \)
  • \( \log(ab) = \log a + \log b \)
  • Trigonometric identities (e.g., \( \sin^2x + \cos^2x = 1 \))

Examples

1. Prove that \( (x + y)^2 – (x – y)^2 = 4xy \)

2. Prove that \( \frac{x^2 – 1}{x – 1} = x + 1 \), for \( x \ne 1 \)

▶️Answer/Explanation

1. LHS: \( (x + y)^2 – (x – y)^2 \)
Expand both squares:
\( = x^2 + 2xy + y^2 – (x^2 – 2xy + y^2) \)
Remove brackets:
\( = x^2 + 2xy + y^2 – x^2 + 2xy – y^2 \)
Simplify:
\( = 4xy \)
RHS = 4xy ⇒ LHS = RHS 

2. LHS: \( \frac{x^2 – 1}{x – 1} \)
Recognize difference of squares:
\( = \frac{(x – 1)(x + 1)}{x – 1} \)
Cancel common factor (since \( x \ne 1 \)):
\( = x + 1 \)
RHS = x + 1 ⇒ LHS = RHS 

Symbols and Notation for Equality and Identity

Symbols and Notation for Equality and Identity

In mathematics, different symbols are used to indicate relationships between expressions. Two of the most important are:

    • \( = \) – Equality symbol

Means the two expressions have the same value for specific inputs or conditions. It is used in equations.

    • \( \equiv \) – Identity symbol

Means the two expressions are always equal for all values in the domain. Used to show that an equation is true by definition or algebraically for all inputs.

Key Distinction:

Equality (\( = \)) can depend on specific values or conditions.
Identity (\( \equiv \)) is always true, regardless of the value of the variable.

Examples

1. \( 2x + 3 = 7 \)

2. \( (x + 1)^2 \equiv x^2 + 2x + 1 \)

▶️ Answer/Explanation

1. \( 2x + 3 = 7 \) is an equation.
It is true only when \( x = 2 \). This is a conditional equality.

2. \( (x + 1)^2 \equiv x^2 + 2x + 1 \) is an identity.
It holds for all values of \( x \), as it is algebraically true by expansion.

Example

Show that \( (x – 3)^2 + 5 \equiv x^2 – 6x + 14 \)

▶️ Answer/Explanation

Start with the LHS: \( (x – 3)^2 + 5 \)

Expand the square: \( x^2 – 6x + 9 + 5 \)

Simplify: \( x^2 – 6x + 14 \)

This matches the RHS exactly. 

Conclusion: \( \text{LHS} \equiv \text{RHS} \), identity proven.

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