</section> <section style="font-size: 50px";> <h2 id="logarithms">Logarithms</h2> <ul> <li>Definition of logarithms</li> <li>Properties of logarithms</li> <li>Common logarithms and natural logarithms</li> <li>Logarithmic equations</li> <li>Logarithmic inequalities

</section> <section style="font-size: 50px";> <h2 id="definition-of-logarithms">Definition of Logarithms</h2> <ul> <li>Definition of a logarithm</li> <li>Relation between exponential and logarithmic functions</li> <li>Logarithmic notation: log_b(x)</li> <li>Base b in logarithmic equations</li> <li>Domain and range of logarithmic functions</li> </ul>

</section> <section style="font-size: 50px";> <h2 id="properties-of-logarithms">Properties of Logarithms</h2> <ul> <li>Product rule: log_b(xy) = log_b(x) + log_b(y)</li> <li>Quotient rule: log_b(x/y) = log_b(x) - log_b(y)</li> <li>Power rule: log_b(xⁿ) = n * log_b(x)</li> <li>Change of base formula: log_b(x) = log_c(x) / log_c(b)

</section> <section style="font-size: 50px";> <h2 id="common-logarithms-and-natural-logarithms">Common Logarithms and Natural Logarithms</h2> <ul> <li>Common logarithm: log(x), base 10 logarithm</li> <li>Natural logarithm: ln(x), base e logarithm</li> <li>ln(x) as the inverse of e^(x)</li> <li>Evaluating logarithms using calculators

</section> <section style="font-size: 50px";> <h2 id="logarithmic-equations">Logarithmic Equations</h2> <ul> <li>Solving logarithmic equations</li> <li>Isolating the logarithm on one side</li> <li>Changing the equation into exponential form</li> <li>Solving for the variable Example: Solve log₄(x-1) = log₂(x-3)

</section> <section style="font-size: 50px";> <h2 id="logarithmic-inequalities">Logarithmic Inequalities</h2> <ul> <li>Solving logarithmic inequalities</li> <li>Differentiating between logarithmic equations and inequalities</li> <li>Interval notation for solutions</li> <li>Graphical representation of logarithmic inequalities</li> </ul>

</section> <section style="font-size: 50px";> <h2 id="exponential-functions">Exponential Functions</h2> <ul> <li>Definition of exponential functions</li> <li>Graphs of exponential functions</li> <li>Exponential growth and decay</li> <li>Exponential equations</li> <li>Solving exponential equations</li> </ul>

</section> <section style="font-size: 50px";> <h2 id="exponential-growth-and-decay">Exponential Growth and Decay</h2> <ul> <li>Exponential growth: y = a * b^x (b > 1)</li> <li>Exponential decay: y = a * b^x (0 < b < 1)</li> <li>Finding growth/decay factors and initial values</li> <li>Applications of exponential growth and decay</li> </ul>

</section> <section style="font-size: 50px";> <h2 id="exponential-equations">Exponential Equations</h2> <ul> <li>Solving exponential equations</li> <li>Using logarithms to solve exponential equations</li> <li>Logarithmic properties for solving exponential equations</li> <li>Common types of exponential equations</li> </ul>

</section> <section style="font-size: 50px";> <h2 id="logarithmic---example--solve-for-x">Logarithmic - Example – Solve for x</h2> <p>Given: log₄(x-1) = log₂(x-3) To solve this equation, we can set the expressions inside the logarithms equal to each other: x - 1 = x - 3 Simplifying, we find:</p> </section> <section style="font-size: 50px";> <p>4 = 2 Since the equation does not hold true, there is no solution for x in this case.

</section> <section style="font-size: 50px";> <h2 id="trigonometry">Trigonometry</h2> <ul> <li>Definition of trigonometric functions</li> <li>Sine, cosine, and tangent functions</li> <li>Trigonometric ratios</li> <li>Unit circle and trigonometric values</li> <li>Trigonometric identities

</section> <section style="font-size: 50px";> <h2 id="trigonometric-functions">Trigonometric Functions</h2> <ul> <li>Evaluating trigonometric functions</li> <li>Special angles and their values</li> <li>Using reference angles</li> <li>Periodicity of trigonometric functions</li> <li>Trigonometric functions on the unit circle Example: Find sin(45 degrees)

</section> <section style="font-size: 50px";> <h2 id="trigonometric-identities">Trigonometric Identities</h2> <ul> <li>Pythagorean identity: sin^2(x) + cos^2(x) = 1</li> <li>Reciprocal identities: csc(x) = 1/sin(x), sec(x) = 1/cos(x), cot(x) = 1/tan(x)</li> <li>Quotient identities: tan(x) = sin(x)/cos(x), cot(x) = cos(x)/sin(x)</li> <li>Additional identities: sin(2x) = 2sin(x)cos(x), cos(2x) = cos^2(x) - sin^2(x) Example: Prove the identity tan^2(x) + 1 = sec^2(x)

</section> <section style="font-size: 50px";> <h2 id="trigonometric-equations">Trigonometric Equations</h2> <ul> <li>Solving trigonometric equations</li> <li>Using algebraic techniques to solve equations</li> <li>Applying trigonometric identities to simplify equations</li> <li>Checking for extraneous solutions</li> <li>Multiple solutions to trigonometric equations Example: Solve sin(x) = cos(x)

</section> <section style="font-size: 50px";> <h2 id="complex-numbers">Complex Numbers</h2> <ul> <li>Definition of complex numbers</li> <li>Imaginary unit: i = sqrt(-1)</li> <li>Complex numbers in rectangular form</li> <li>Complex numbers in polar form</li> <li>Operations with complex numbers

</section> <section style="font-size: 50px";> <h2 id="rectangular-and-polar-forms">Rectangular and Polar Forms</h2> <ul> <li>Converting from rectangular to polar form</li> <li>Finding the magnitude and argument of a complex number</li> <li>Converting from polar to rectangular form</li> <li>Euler’s formula: e^(ix) = cos(x) + isin(x)</li> <li>De Moivre’s theorem: (cos(x) + isin(x))^n = cos(nx) + isin(nx) Example: Convert 3 + 4i to polar form

</section> <section style="font-size: 50px";> <h2 id="operations-with-complex-numbers">Operations with Complex Numbers</h2> <ul> <li>Addition and subtraction of complex numbers</li> <li>Multiplication and division of complex numbers</li> <li>Properties of complex numbers</li> <li>Complex conjugates and their properties</li> <li>Powers and roots of complex numbers Example: Simplify (2 + i)(3 - 2i)

</section> <section style="font-size: 50px";> <h2 id="sequences-and-series">Sequences and Series</h2> <ul> <li>Definition of sequences and series</li> <li>Arithmetic sequences and series</li> <li>Geometric sequences and series</li> <li>Formulas for the nth term and sum of each type</li> <li>Sums of infinite geometric series Example: Find the sum of the arithmetic series 2 + 5 + 8 + … + 23

</section> <section style="font-size: 50px";> <h2 id="arithmetic-sequences">Arithmetic Sequences</h2> <ul> <li>Defining arithmetic sequences</li> <li>Formula for the nth term of an arithmetic sequence</li> <li>Formula for the sum of an arithmetic sequence</li> <li>Solving problems involving arithmetic sequences</li> <li>Applications of arithmetic sequences in real life Example: Find the common difference of an arithmetic sequence if the first term is 5 and the fifth term is 20

</section> <section style="font-size: 50px";> <h2 id="geometric-sequences">Geometric Sequences</h2> <ul> <li>Defining geometric sequences</li> <li>Formula for the nth term of a geometric sequence</li> <li>Formula for the sum of a geometric series</li> <li>Solving problems involving geometric sequences</li> <li>Applications of geometric sequences in real life Example: Find the sum of the geometric series 3 + 9 + 27 + … + 6561

</section> <section style="font-size: 50px";> <h2 id="matrices">Matrices</h2> <ul> <li>Definition and notation of matrices</li> <li>Types of matrices: square, rectangular, diagonal, identity, zero</li> <li>Matrix operations: addition, subtraction, scalar multiplication</li> <li>Matrix multiplication: properties and rules</li> <li>Transpose of a matrix Example: Multiply the matrices A = [1 2; 3 4] and B = [5 6; 7 8]

</section> <section style="font-size: 50px";> <h2 id="determinants">Determinants</h2> <ul> <li>Definition of determinants</li> <li>Properties of determinants</li> <li>Calculating determinants of 2x2 and 3x3 matrices</li> <li>Cofactor expansion method</li> <li>Inverse of a matrix using determinants Example: Find the determinant of the matrix A = [2 3; 4 5]

</section> <section style="font-size: 50px";> <h2 id="systems-of-equations">Systems of Equations</h2> <ul> <li>Solving systems of equations</li> <li>Types of solutions: consistent, inconsistent, dependent</li> <li>Methods for solving: substitution, elimination, matrices</li> <li>Solving systems with three variables</li> <li>Applications of systems of equations Example: Solve the system of equations: 2x + y = 5 and 3x - 2y = 8

</section> <section style="font-size: 50px";> <h2 id="probability">Probability</h2> <ul> <li>Basics of probability: sample space, events, outcomes</li> <li>Probability notation and rules: addition, multiplication</li> <li>Conditional probability and independence</li> <li>Permutations and combinations</li> <li>Probability distributions Example: A bag contains 5 red and 3 blue marbles. What is the probability of drawing a red marble?

</section> <section style="font-size: 50px";> <h2 id="statistics">Statistics</h2> <ul> <li>Measures of central tendency: mean, median, mode</li> <li>Measures of dispersion: range, variance, standard deviation</li> <li>Data representation: bar graphs, histograms, box plots</li> <li>Correlation and regression analysis</li> <li>Hypothesis testing Example: Calculate the mean, median, and mode for the dataset: 4, 6, 2, 8, 5

</section> <section style="font-size: 50px";> <h2 id="calculus">Calculus</h2> <ul> <li>Limits and continuity</li> <li>Differentiation: rules, chain rule, implicit differentiation</li> <li>Applications of differentiation: rates of change, optimization</li> <li>Integration: rules, definite and indefinite integrals</li> <li>Applications of integration: area under the curve, volumes of revolution Example: Find the derivative of the function f(x) = 3x^2 - 2x + 1

</section> <section style="font-size: 50px";> <h2 id="vectors">Vectors</h2> <ul> <li>Definition and notation of vectors</li> <li>Vector operations: addition, scalar multiplication</li> <li>Dot product and cross product</li> <li>Vector components and magnitude</li> <li>Applications of vectors in physics and geometry Example: Find the dot product of vectors A = 2i + 3j and B = -4i + 5j

</section> <section style="font-size: 50px";> <h2 id="complex-analysis">Complex Analysis</h2> <ul> <li>Complex functions and mappings</li> <li>Complex differentiation and integration</li> <li>Complex roots and residues</li> <li>Applications of complex analysis in physics and engineering</li> <li>Complex analysis in graphical representation Example: Find the square root of -9 using complex numbers

</section> <section style="font-size: 50px";> <h2 id="differential-equations">Differential Equations</h2> <ul> <li>Definitions and classifications of differential equations</li> <li>Solution methods: separation of variables, integrating factor, substitution</li> <li>First-order and higher-order differential equations</li> <li>Applications of differential equations in science and engineering</li> <li>Systems of differential equations Example: Solve the differential equation dy/dx = 2x

</section> <section style="font-size: 50px";> <h2 id="3d-geometry">3D Geometry</h2> <ul> <li>Cartesian coordinates in 3D space</li> <li>Equations of lines and planes in 3D</li> <li>Distance and midpoint formulas</li> <li>Vector representations in 3D</li> <li>Intersection of lines and planes in 3D Example: Find the equation of the plane passing through the points (1, 2, 3), (2, 4, 6), and (3, 6, 9)