Topic: Logarithm

• Definition of logarithm
• Logarithm properties
  • Product rule
  • Quotient rule
  • Power rule

Definition of Logarithm

• The logarithm of a number with respect to a given base is the exponent to which the base must be raised to obtain that number.
• Example: log_a(x) = y ⟺ a^y = x

Logarithm Properties

Product Rule

• log_a(xy) = log_a(x) + log_a(y)

Quotient Rule

• log_a(x/y) = log_a(x) - log_a(y)

Power Rule

• log_a(xⁿ) = n * log_a(x)

Topic: Logarithm - Example

• Solve the equation: log_a(x+3) + log_a(x-3) = log_a(16)

Solution: Logarithm - Example

1. Use the product rule to simplify the equation:
   • log_a((x+3)(x-3)) = log_a(16)
     
   • log_a(x²-9) = log_a(16)

2. Apply the exponentiation property to remove the logarithm:
   • x²-9 = 16

Solution: Logarithm - Example (continued)

3. Rearrange the equation to solve for x:
   • x² = 25
     • x = ±5

4. Check if the obtained solution satisfies the initial equation:
   • log_a(5+3) + log_a(5-3) = log_a(16)
   • log_a(8) + log_a(2) = log_a(16)

Logarithm - Example (continued)

5. Evaluate the logarithms using the logarithm properties:
   • log_a(8) + log_a(2) = log_a(16)
     • 3 + 1 = 4
     • 4 = 4

6. The obtained solution x = 5 satisfies the given equation.

Topic: Complex Numbers

• Introduction to complex numbers
• Imaginary unit (i)
• Real and imaginary parts of a complex number
• Conjugate of a complex number

Introduction to Complex Numbers

• A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit.

Imaginary Unit (i)

• The imaginary unit (i) is defined as the square root of -1.
• i² = -1

Real and Imaginary Parts of a Complex Number

• The real part of a complex number a+bi is the real number a.
• The imaginary part of a complex number a+bi is the real number b.
• Example: For the complex number 3+4i, the real part is 3 and the imaginary part is 4.

Conjugate of a Complex Number

• The conjugate of a complex number a+bi is denoted as a-bi.
• The conjugate of a complex number has the same real part but the opposite sign for the imaginary part.
• Example: The conjugate of the complex number 5+2i is 5-2i.

Topic: Complex Numbers - Example

• Simplify the expression: (3+2i)(5-4i)

Solution: Complex Numbers - Example

1. Use the distributive property of multiplication to expand the expression:
   • (3+2i)(5-4i) = 3(5) + 3(-4i) + 2i(5) + 2i(-4i)

2. Simplify each term:
   • 15 - 12i + 10i - 8i²

Solution: Complex Numbers - Example (continued)

3. Use the fact that i² = -1 to simplify further:
   • 15 - 12i + 10i + 8

4. Combine like terms:
   • 23 - 2i

5. The simplified expression is 23 - 2i.

Topic: Matrices

• Definition of a matrix
• Order (or dimensions) of a matrix
• Types of matrices: square, rectangular, identity, zero
• Addition and subtraction of matrices

Definition of a Matrix

• A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns.

Order of a Matrix

• The order (or dimensions) of a matrix is defined by the number of rows and columns it has.
• The order is represented as (m x n), where m represents the number of rows and n represents the number of columns.

Types of Matrices

• Square Matrix: A matrix with an equal number of rows and columns.
• Rectangular Matrix: A matrix with a different number of rows and columns.
• Identity Matrix: A square matrix with ones on the main diagonal and zeros elsewhere.
• Zero Matrix: A matrix with all elements equal to zero.

Addition and Subtraction of Matrices

• Matrices can be added or subtracted when they have the same dimensions.
• To add or subtract matrices, simply add or subtract corresponding elements.
• Example: For matrices A = [1 2 3] and B = [4 5 6], A + B = [5 7 9]

Topic: Logarithm - Example

• Solve for x: log_a(x+3) + log_a(x-3) = log_a(16)
• Use the product rule of logarithms: log_a(x+3)(x-3) = log_a(16)
• Simplify the equation: log_a(x²-9) = log_a(16)
• Apply the exponentiation property: x² - 9 = 16
• Rearrange the equation: x² = 25

Solution: Logarithm - Example

• Solve for x: x² = 25
• Take the square root of both sides: x = ±5
• Check the solution: log_a(5+3) + log_a(5-3) = log_a(16)
• Simplify each term: log_a(8) + log_a(2) = log_a(16)
• Apply the logarithm properties: 3 + 1 = 4

Topic: Complex Numbers - Example

• Simplify the expression: (3+2i)(5-4i)
• Use the distributive property: 3(5) + 3(-4i) + 2i(5) + 2i(-4i)
• Simplify each term: 15 - 12i + 10i - 8i^2
• Apply the fact that i^2 = -1: 15 - 12i + 10i + 8

Solution: Complex Numbers - Example

• Combine like terms: 23 - 2i
• The simplified expression is 23 - 2i
• Note: In complex number multiplication, the terms i and i^2 can be combined to simplify the expression.
• Remember: i^2 = -1

Topic: Matrices

• Definition of a matrix: a rectangular array of numbers, symbols, or expressions arranged in rows and columns
• Order (or dimensions) of a matrix: represented as (m x n), where m is the number of rows and n is the number of columns
• Types of matrices: square, rectangular, identity, zero
• Addition and subtraction of matrices: possible when the matrices have the same dimensions

Order of a Matrix

• The order of a matrix is defined by the number of rows and columns it has.
• Matrix A with m rows and n columns: written as A_{m x n}
• Example: Matrix A with 3 rows and 2 columns is written as A_{3 x 2}
• Rows are horizontal and columns are vertical

Types of Matrices

• Square Matrix: a matrix with an equal number of rows and columns
• Rectangular Matrix: a matrix with a different number of rows and columns
• Identity Matrix: a square matrix with ones on the main diagonal and zeros elsewhere
• Zero Matrix: a matrix with all elements equal to zero
• Example: Identity matrix with order 3: I₃ = {{1, 0, 0}, {0, 1, 0}, {0, 0, 1}}

Addition and Subtraction of Matrices

• Matrices can be added or subtracted when they have the same dimensions.
• To add or subtract matrices, simply add or subtract corresponding elements.
• Example: For matrices A = {{1, 2, 3}} and B = {{4, 5, 6}}, A + B = {{5, 7, 9}}
• Example: For matrices C = {{5, 6, 7}} and D = {{1, 2, 3}}, C - D = {{4, 4, 4}}

Applications of Matrices

• Matrices are extensively used in various fields such as physics, computer science, economics, and engineering.
• They can represent linear transformations, solve systems of linear equations, analyze networks, and perform operations on data sets.
• Matrices are also used in optimization problems, cryptography, and image processing.

Summary

• Logarithms are powerful tools for solving equations involving exponents.
• Complex numbers combine real and imaginary parts and can be simplified using algebraic operations.
• Matrices are rectangular arrays of numbers used in various applications and can be added or subtracted when they have the same dimensions.
• Understanding these topics will be beneficial for your 12th Boards Math exam and future mathematical applications.