Logarithm

Definition:
- The logarithm of a number to a given base is the exponent to which the base must be raised to produce that number.
- Example: $\log_{2}8 = 3$ because $2^{3} = 8$
Properties of Logarithms:
- $\log_{b}(xy) = \log_{b}x + \log_{b}y$
- $\log_{b}\left(\frac{x}{y}\right) = \log_{b}x - \log_{b}y$
- $\log_{b}(x^{n}) = n\log_{b}x$
- $\log_{b}1 = 0$
- $\log_{b}b = 1$
Example 1:
- Simplify $\log_{3}27$
- Solution: Since $3^{3} = 27$ , the logarithm of 27 to the base 3 is 3. Therefore, $\log_{3}27 = 3$
Example 2:
- Solve for $x$ : $\log_{5}x = 2$
- Solution: Since $5^{2} = 25$ , the value of $x$ is 25. Therefore, $x = 25$

Exponents and Radicals

Laws of Exponents:
- Product Rule: $a^{m} \times a^{n} = a^{m+n}$
- Quotient Rule: $\frac{a^{m}}{a^{n}} = a^{m-n}$
- Power Rule: $(a^{m})^{n} = a^{mn}$
Laws of Radicals:
- Product Rule: $\sqrt[a]{bc} = \sqrt[a]{b} \cdot \sqrt[a]{c}$
- Quotient Rule: $\frac{\sqrt[a]{b}}{\sqrt[a]{c}} = \sqrt[a]{\frac{b}{c}}$
- Power Rule: $(\sqrt[a]{b})^{c} = \sqrt[a]{b^{c}}$
Example 1:
- Simplify: $2^{3} \times 2^{2}$
- Solution: Applying the product rule of exponents, we get $2^{3} \times 2^{2} = 2^{3+2} = 2^{5} = 32$
Example 2:
- Simplify: $\sqrt[3]{8} \cdot \sqrt[3]{27}$
- Solution: Applying the product rule of radicals, we get $\sqrt[3]{8} \cdot \sqrt[3]{27} = \sqrt[3]{8 \cdot 27} = \sqrt[3]{216} = 6$

Complex Numbers

Definition:
- A complex number is a number of the form $a + bi$ , where $a$ and $b$ are real numbers and $i$ represents the imaginary unit.
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: $(a + bi) \cdot (c + di) = (ac - bd) + (ad + bc)i$
- Division: $\frac{a + bi}{c + di} = \frac{(ac + bd) - (bc - ad)i}{c^{2} + d^{2}}$
Example 1:
- Add: $(3 + 4i) + (2 + 5i)$
- Solution: Adding the real and imaginary parts separately, we get $(3 + 2) + (4 + 5)i = 5 + 9i$
Example 2:
- Multiply: $(3 + 2i) \cdot (4 + 5i)$
- Solution: Applying the distributive property, we get $12 + 15i + 8i + 10i^{2} = 12 + 23i - 10 = 2 + 23i$

Differentiation

Definition:
- Differentiation is the process of finding the derivative of a function.
Rules of Differentiation:
- Constant Rule: $\frac{d}{dx}(c) = 0$ , where $c$ is a constant
- Power Rule: $\frac{d}{dx}(x^{n}) = nx^{n-1}$ , where $n$ is a constant
- Sum/Difference Rule: $\frac{d}{dx}(f(x) \pm g(x)) = \frac{d}{dx}(f(x)) \pm \frac{d}{dx}(g(x))$
- Product Rule: $\frac{d}{dx}(f(x) \cdot g(x)) = f’(x) \cdot g(x) + f(x) \cdot g’(x)$
- Quotient Rule: $\frac{d}{dx}\left(\frac{f(x)}{g(x)}\right) = \frac{f’(x) \cdot g(x) - f(x) \cdot g’(x)}{(g(x))^2}$
Example 1:
- Find the derivative of $f(x) = 3x^2 - 5x + 2$
- Solution: Applying the power rule, we get $f’(x) = 2 \cdot 3x^{2-1} - 1 \cdot 5x^{1-1} = 6x - 5$
Example 2:
- Find the derivative of $f(x) = \frac{x^2}{e^x}$
- Solution: Applying the quotient rule, we get $f’(x) = \frac{(2x)(e^x) - (x^2)(e^x)}{(e^x)^2}$

Integration

Definition:
- Integration is the process of finding the antiderivative of a function.
Techniques of Integration:
- Power Rule: $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ , where $n$ is a constant and $C$ is the constant of integration
- Constant Multiple Rule: $\int cf(x) dx = c \int f(x) dx$ , where $c$ is a constant
- Sum/Difference Rule: $\int (f(x) \pm g(x)) dx = \int f(x) dx \pm \int g(x) dx$
- Substitution Rule: $\int f(g(x)) \cdot g’(x) dx = F(g(x)) + C$ , where $F(x)$ is the antiderivative of $f(x)$ and $g’(x)$ is the derivative of $g(x)$
Example 1:
- Evaluate $\int 3x^2 dx$
- Solution: Applying the power rule, we get $\int 3x^2 dx = \frac{3x^{2+1}}{2+1} + C = x^3 + C$
Example 2:
- Evaluate $\int 2e^{3x} dx$
- Solution: Applying the substitution rule with $u = 3x$ and $du = 3 dx$ , we get $\int 2e^u \cdot \frac{du}{3} = \frac{2}{3} \int e^u du = \frac{2}{3} e^u + C = \frac{2}{3} e^{3x} + C$

Trigonometry

Definitions:
- Sine: In a right triangle, the sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse.
- Cosine: In a right triangle, the cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse.
- Tangent: In a right triangle, the tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side.
Trigonometric Identities:
- Pythagorean Identity: $\sin^2(x) + \cos^2(x) = 1$
- Cofunction Identities: $\sin(\frac{\pi}{2} - x) = \cos(x)$ , $\cos(\frac{\pi}{2} - x) = \sin(x)$ , $\tan(\frac{\pi}{2} - x) = \cot(x)$
- Double Angle Identities: $\sin(2x) = 2\sin(x)\cos(x)$ , $\cos(2x) = \cos^2(x) - \sin^2(x)$ , $\tan(2x) = \frac{2\tan(x)}{1-\tan^2(x)}$
Example 1:
- Find the value of $\sin(\frac{\pi}{6})$
- Solution: Since $\frac{\pi}{6}$ is a special angle, we know that $\sin(\frac{\pi}{6}) = \frac{1}{2}$
Example 2:
- Simplify $\cos\left(\frac{\pi}{2} - \frac{\pi}{3}\right)$
- Solution: Using the cofunction identity, we have $\cos\left(\frac{\pi}{2} - \frac{\pi}{3}\right) = \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}$

Probability

Definitions:
- Probability: The measure of the likelihood that an event will occur.
- Sample Space: The set of all possible outcomes of an experiment.
- Event: A subset of the sample space.
- Probability of an Event: The ratio of the number of favorable outcomes to the number of possible outcomes.
Probability Rules:
- Rule of Complement: $P(A’) = 1 - P(A)$
- Addition Rule: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$
- Multiplication Rule: $P(A \cap B) = P(A) \cdot P(B|A)$ , where $P(B|A)$ is the probability of event B occurring given that event A has already occurred.
Example 1:
- A fair die is rolled. What is the probability of rolling an even number?
- Solution: There are 3 favorable outcomes (2, 4, 6) out of 6 possible outcomes (1, 2, 3, 4, 5, 6). Therefore, the probability is $\frac{3}{6} = \frac{1}{2}$
Example 2:
- Two cards are drawn successively from a standard deck of playing cards without replacement. What is the probability of drawing a heart on the first card and a diamond on the second card?
- Solution: The probability of drawing a heart on the first card is $\frac{13}{52}$ . Since one card has been removed, the probability of drawing a diamond on the second card is $\frac{13}{51}$ (there are now 13 diamonds left out of 51 cards). Therefore, the probability is $\frac{13}{52} \cdot \frac{13}{51}$

Matrices

Definition:
- A matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns.
Matrix Operations:
- Addition: To add two matrices, the corresponding elements are added.
- Subtraction: To subtract two matrices, the corresponding elements are subtracted.
- Scalar Multiplication: To multiply a matrix by a scalar, each element in the matrix is multiplied by the scalar.
- Matrix Multiplication: To multiply two matrices, the rows of the first matrix are multiplied by the columns of the second matrix and the products are summed.
Example 1:
- Add the following matrices:
- $\begin{bmatrix} 2 & 4 \ 1 & 3 \end{bmatrix} + \begin{bmatrix} -1 & 3 \ 2 & 5 \end{bmatrix}$
- Solution: Adding the corresponding elements, we get $\begin{bmatrix} 1 & 7 \ 3 & 8 \end{bmatrix}$
Example 2:
- Multiply the following matrices:
- $\begin{bmatrix} 2 & 3 \ 4 & 1 \end{bmatrix} \cdot \begin{bmatrix} 1 & -2 \ 3 & 5 \end{bmatrix}$
- Solution: Applying matrix multiplication, we get $\begin{bmatrix} 11 & 4 \ 7 & 18 \end{bmatrix}$

Complex Numbers

Definition:
- A complex number is a number of the form $a + bi$ , where $a$ and $b$ are real numbers and $i$ represents the imaginary unit.
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: $(a + bi) \cdot (c + di) = (ac - bd) + (ad + bc)i$
- Division: $\frac{a + bi}{c + di} = \frac{(ac + bd) - (bc - ad)i}{c^{2} + d^{2}}$
Example 1:
- Add: $(3 + 4i) + (2 + 5i)$
- Solution: Adding the real and imaginary parts separately, we get $(3 + 2) + (4 + 5)i = 5 + 9i$
Example 2:
- Multiply: $(3 + 2i) \cdot (4 + 5i)$
- Solution: Applying the distributive property, we get $12 + 15i + 8i + 10i^{2} = 12 + 23i - 10 = 2 + 23i$

Vectors

Definition:
- A vector is a quantity that has both magnitude and direction.
Vector Operations:
- Addition: To add two vectors, the corresponding components are added.
- Subtraction: To subtract two vectors, the corresponding components are subtracted.
- Scalar Multiplication: To multiply a vector by a scalar, each component in the vector is multiplied by the scalar.
- Dot Product: To find the dot product of two vectors, the corresponding components are multiplied and then summed.
- Cross Product: To find the cross product of two vectors, the magnitude and direction are calculated according to specific rules.
Example 1:
- Add the following vectors:
- $\vec{v} = \begin{bmatrix} 2 \ 5 \end{bmatrix}$ and $\vec{u} = \begin{bmatrix} -1 \ 3 \end{bmatrix}$
- Solution: Adding the corresponding components, we get $\vec{v} + \vec{u} = \begin{bmatrix} 2 + (-1) \ 5 + 3 \end{bmatrix}$
Example 2:
- Find the dot product of the following vectors:
- $\vec{v} = \begin{bmatrix} 2 \ -3 \end{bmatrix}$ and $\vec{u} = \begin{bmatrix} 4 \ 1 \end{bmatrix}$
- Solution: Multiplying the corresponding components and then summing, we get $\vec{v} \cdot \vec{u} = (2)(4) + (-3)(1)$

Logarithm

Exponents and Radicals

Complex Numbers

Differentiation

Integration

Trigonometry

Probability

Matrices

Complex Numbers

Vectors

Logarithm - Example – Properties of logarithms

Exponents and Radicals - Example – Laws of exponents

Complex Numbers - Example – Addition and multiplication of complex numbers