Logarithm

  • Definition: The logarithm function $\log_b x$ gives you the exponent to which the base $b$ must be raised to obtain the number $x$.
  • Properties:
    • $\log_b (xy) = \log_b x + \log_b y$
    • $\log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y$
    • $\log_b x^n = n \cdot \log_b x$
    • $\log_b b = 1$
    • $\log_b 1 = 0$

Logarithmic Equations

  • Logarithmic equations are equations that involve logarithms.
  • Examples:
    1. Solve for $x$: $\log_2 x = 3$
    2. Solve for $x$: $\log_5 (2x-4) = 2$

Inequalities

  • Inequalities are mathematical expressions involving greater than ($>$), less than ($<$), greater than or equal to ($\geq$), less than or equal to ($\leq$), etc.
  • Properties:
    • When multiplying or dividing by a negative number, the direction of the inequality sign flips.
    • When we multiply or divide a compound inequality by a negative number, we flip the direction of the inequality sign.

Inequality Examples

  1. Solve for $x$: $3x + 5 > 10$
  1. Solve for $x$: $\frac{x+3}{4} < 2$

Quadratic Equations

  • Quadratic equations are equations of the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants and $a \neq 0$.
  • Solving methods:
    1. Factoring
    2. Quadratic formula
    3. Completing the square

Solving Quadratic Equations by Factoring

  • Examples:
    1. Solve for $x$: $x^2 + 5x + 6 = 0$
    2. Solve for $x$: $3x^2 - 2x - 1 = 0$

Quadratic Formula

  • The quadratic formula is used to solve any quadratic equation: $ax^2 + bx + c = 0$.
  • The formula is given by: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
  • Examples:
    1. Solve for $x$: $2x^2 + 3x - 2 = 0$
    2. Solve for $x$: $x^2 - 7x + 12 = 0$

Completing the Square

  • Steps:
    1. Write the quadratic equation in the form $ax^2 + bx + c = 0$.
    2. Determine the value of $c$ by rearranging the equation.
    3. Complete the square by adding $(\frac{b}{2})^2$ to both sides of the equation.
    4. Factorize the perfect square trinomial and solve for $x$.
  • Examples:
    1. Solve for $x$: $x^2 - 6x + 8 = 0$
    2. Solve for $x$: $2x^2 + 4x - 6 = 0$

Complex Numbers

  • Complex numbers are numbers in the form $a + bi$, where $a$ and $b$ are real numbers.
  • The imaginary unit $i$ is defined as $\sqrt{-1}$.
  • Operations with complex numbers:
    • Addition: $(a + bi) + (c + di) = (a + c) + (b + d)i$
    • Subtraction: $(a + bi) - (c + di) = (a - c) + (b - d)i$
    • Multiplication: $(a + bi)(c + di) = (ac - bd) + (ad + bc)i$
    • Division: $\frac{a + bi}{c + di} = \frac{(ac + bd) + (bc - ad)i}{c^2 + d^2}$

Complex Conjugates

  • The complex conjugate of a complex number $a + bi$ is $a - bi$.
  • Properties:
    • The product of a complex number and its conjugate is a real number: $(a + bi)(a - bi) = a^2 + b^2$.
    • Dividing a complex number by its conjugate gives a real number: $\frac{a + bi}{a - bi} = \frac{a^2 + b^2}{a^2 + b^2} = 1$

Logarithm - 3 Inequality for n

  • Logarithm inequality for n is an inequality involving logarithmic expressions.
  • Example: Solve for $n$: $\log_2 (n + 3) - \log_2 (n - 2) > 1$

Logarithm - 3 Inequality Solution

  • Steps to solve logarithm inequality for n:
    1. Rewrite the inequality: $\log_2 \left(\frac{n + 3}{n - 2}\right) > 1$
    2. Express 1 as a logarithmic expression: $\log_2 2 > \log_2 \left(\frac{n + 3}{n - 2}\right)$
    3. Simplify: $1 > \log_2 \left(\frac{n + 3}{n - 2}\right)$
    4. Rewrite as an exponential equation: $2^1 > \frac{n + 3}{n - 2}$
    5. Solve for n: $2 > \frac{n + 3}{n - 2}$

Logarithm - 3 Inequality Solution (cont.)

  • Steps to solve logarithm inequality for n (cont.): 6. Cross-multiply: $2(n - 2) > n + 3$ 7. Simplify: $2n - 4 > n + 3$ 8. Combine like terms: $n > 7$

Inequality for n - 4

  • Inequality for n is an inequality involving variable n.
  • Example: Solve for n: $3n - 4 \geq 10$

Inequality Solution for n - 4

  • Steps to solve inequality for n:
    1. Add 4 to both sides: $3n - 4 + 4 \geq 10 + 4$
    2. Simplify: $3n \geq 14$
    3. Divide both sides by 3: $\frac{3n}{3} \geq \frac{14}{3}$

Inequality Solution for n - 4 (cont.)

  • Steps to solve inequality for n (cont.): 4. Simplify: $n \geq \frac{14}{3}$

Quadratic Equations - Example 1

  • Example: Solve for x: $2x^2 - 3x - 2 = 0$
  • Using quadratic formula:
    • $x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(2)(-2)}}{2(2)}$
    • Simplify: $x = \frac{3 \pm \sqrt{9 + 16}}{4}$
    • Simplify: $x = \frac{3 \pm \sqrt{25}}{4}$
    • Simplify: $x = \frac{3 \pm 5}{4}$
    • Solutions: $x = \frac{8}{4} = 2$ or $x = \frac{-2}{4} = -\frac{1}{2}$

Quadratic Equations - Example 2

  • Example: Solve for x: $x^2 + 5x + 6 = 0$
  • Factoring: $(x + 2)(x + 3) = 0$
  • Set each factor equal to 0: $x + 2 = 0$ or $x + 3 = 0$
  • Solve for x: $x = -2$ or $x = -3$

Complex Numbers - Addition

  • Addition of complex numbers:
    • $(a + bi) + (c + di) = (a + c) + (b + d)i$
  • Example: $(4 + 3i) + (2 + 5i)$
  • Simplify: $6 + 8i$

Complex Numbers - Multiplication

  • Multiplication of complex numbers:
    • $(a + bi)(c + di) = (ac - bd) + (ad + bc)i$
  • Example: $(2 + 3i)(4 + i)$
  • Simplify: $8 + 2i + 12i - 3$
  • Simplify: $5 + 14i$

Logarithm - 3 Inequality for n

  • Logarithm inequality for n is an inequality involving logarithmic expressions.
  • Example: Solve for $n$: $\log_2 (n + 3) - \log_2 (n - 2) > 1$
    • Steps:
      1. Rewrite the inequality: $\log_2 \left(\frac{n + 3}{n - 2}\right) > 1$
      2. Express 1 as a logarithmic expression: $\log_2 2 > \log_2 \left(\frac{n + 3}{n - 2}\right)$
      3. Simplify: $1 > \log_2 \left(\frac{n + 3}{n - 2}\right)$
      4. Rewrite as an exponential equation: $2^1 > \frac{n + 3}{n - 2}$
      5. Solve for n: $2 > \frac{n + 3}{n - 2}$

Logarithm - 3 Inequality for n Solution

  • Steps to solve logarithm inequality for n (cont.): 6. Cross-multiply: $2(n - 2) > n + 3$ 7. Simplify: $2n - 4 > n + 3$ 8. Combine like terms: $n > 7$

Inequality for n - 4

  • Inequality for n is an inequality involving variable n.
  • Example: Solve for n: $3n - 4 \geq 10$
    • Steps:
      1. Add 4 to both sides: $3n - 4 + 4 \geq 10 + 4$
      2. Simplify: $3n \geq 14$
      3. Divide both sides by 3: $\frac{3n}{3} \geq \frac{14}{3}$

Inequality for n - 4 Solution

  • Steps to solve inequality for n (cont.): 4. Simplify: $n \geq \frac{14}{3}$

Quadratic Equations - Example 1

  • Example: Solve for x: $2x^2 - 3x - 2 = 0$
  • Using quadratic formula:
    • $x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(2)(-2)}}{2(2)}$
    • Simplify: $x = \frac{3 \pm \sqrt{9 + 16}}{4}$
    • Simplify: $x = \frac{3 \pm \sqrt{25}}{4}$
    • Simplify: $x = \frac{3 \pm 5}{4}$
    • Solutions: $x = \frac{8}{4} = 2$ or $x = \frac{-2}{4} = -\frac{1}{2}$

Quadratic Equations - Example 2

  • Example: Solve for x: $x^2 + 5x + 6 = 0$
  • Factoring: $(x + 2)(x + 3) = 0$
  • Set each factor equal to 0: $x + 2 = 0$ or $x + 3 = 0$
  • Solve for x: $x = -2$ or $x = -3$

Complex Numbers - Addition

  • Addition of complex numbers:
    • $(a + bi) + (c + di) = (a + c) + (b + d)i$
  • Example: $(4 + 3i) + (2 + 5i)$
  • Simplify: $6 + 8i$

Complex Numbers - Multiplication

  • Multiplication of complex numbers:
    • $(a + bi)(c + di) = (ac - bd) + (ad + bc)i$
  • Example: $(2 + 3i)(4 + i)$
  • Simplify: $8 + 2i + 12i - 3$
  • Simplify: $5 + 14i$

Complex Conjugates

  • The complex conjugate of a complex number $a + bi$ is $a - bi$.
  • Properties:
    • The product of a complex number and its conjugate is a real number: $(a + bi)(a - bi) = a^2 + b^2$.
    • Dividing a complex number by its conjugate gives a real number: $\frac{a + bi}{a - bi} = \frac{a^2 + b^2}{a^2 + b^2} = 1$

Inequalities

  • Inequalities are mathematical expressions involving greater than ($>$), less than ($<$), greater than or equal to ($\geq$), less than or equal to ($\leq$), etc.
  • Properties:
    • When multiplying or dividing by a negative number, the direction of the inequality sign flips.
    • When we multiply or divide a compound inequality by a negative number, we flip the direction of the inequality sign.