Exponents and Logs
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This document covers exponents, logarithms, and their functions. It defines exponential and logarithmic expressions, reviews rules for manipulating exponents and logarithms, and examines the graphs of exponential and logarithmic functions. It provides examples of applying exponent and logarithm rules, and contains activities involving evaluating exponential and logarithmic expressions and graphing their related functions.
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(1) The document provides examples of writing basic Matlab programs using variables, arrays, mathematical operations, conditional statements, and loops.
(2) Key concepts covered include assigning values to variables, built-in math functions, formatted output, extracting individual elements from arrays, using comments, and plotting simple graphs.
(3) Examples demonstrate the use of conditional statements like if-elseif-else and logical operators, as well as for loops and while loops to perform repetitive calculations and operations on arrays.
chapter3.ppt
The document discusses exponential and logarithmic functions. It defines exponential functions as functions of the form f(x) = bx, where b is the base. It provides examples of graphs of exponential functions with different bases. It then introduces logarithmic functions as the inverse of exponential functions, where logarithms are defined as logbx = y if x = by. It provides properties and examples involving logarithmic functions.
Logarithmic Functions
The document defines logarithmic functions and provides examples of converting between logarithmic and exponential forms. It discusses the properties of logarithmic equality and solving logarithmic equations by rewriting them in exponential form. Key points include: logarithmic and exponential forms are inverses; the exponent becomes the logarithm; logarithmic functions have a domain of positive real numbers; and solving logarithms involves setting arguments equal when bases are the same.
5indiceslogarithms 120909011915-phpapp02
Logarithms and indices are important mathematical concepts. Laws and formulas allow logarithmic and index expressions to be simplified and equations to be solved.
Key steps in working with logarithms and indices include using the appropriate laws and formulas to simplify expressions, setting up logarithmic or index equations equal to each other, and solving the resulting linear equations.
Practice questions cover a range of skills like evaluating logarithmic expressions without a calculator, expressing logarithmic values in terms of given variables, and solving equations involving logarithms and indices. Mastering the laws, formulas, and problem-solving process is essential for working with logarithms and indices.
Mc ty-logarithms-2009-1
Logarithms are used to replace multiplication with addition and division with subtraction. They allow calculations with large numbers to be simplified. There are three main laws of logarithms: 1) the logarithm of a product is equal to the sum of the logarithms of the factors, 2) the logarithm of a power is the power multiplied by the logarithm of the number, and 3) the logarithm of a quotient is equal to the logarithm of the dividend minus the logarithm of the divisor. Logarithms are commonly written to bases 10 and e, known as common logarithms and natural logarithms.
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This document discusses solenoids and how to determine the magnetic field around them. It explains that a solenoid is a coil of wire that produces a magnetic field similar to a bar magnet when electric current flows through it. The right hand rule is used to determine the north and south poles of a solenoid by placing the hand at the positive end and orienting the thumb to point north. Examples are given to demonstrate how to apply the right hand rule to identify pole orientation in different solenoid setups.
Electromagnets.ppt
This document discusses electromagnets and the factors that affect their magnetic fields. It explains that an electromagnet's magnetic field is produced by electric current flowing through a coil. The key factors that strengthen the magnetic field are:
1. Using a ferromagnetic core material like iron inside the coil
2. Increasing the number of loops or turns of wire in the coil
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The document provides examples comparing different electromagnets based on variations in these factors, and explains that the magnetic force is calculated as the product of the current and number of turns.
Concrete Beam.pptx
This document provides information about concrete beams and materials. It discusses:
- The types of forces and deformations that can act on materials, including compression, tension, torsion, deflection, shearing, elastic, plastic, and fracture.
- The five main categories of materials - wood, ceramics, metals, plastics, and composites.
- Key properties and behaviors of each material category, how they degrade, and how they can be protected.
- Tests that are commonly performed on concrete to ensure it meets specifications, including slump, air content, density, and compressive strength.
Technical Drawing & Assembly.ppt
The document discusses various topics related to manufacturing technical objects including:
- Guiding components which control the motion of moving parts and can provide translational, rotational, or helical motion.
- Degrees of freedom which describe the possible independent movements of parts.
- Technical drawings including projections, general arrangements, exploded views, details, dimensions, and developments.
- Manufacturing processes involving measuring, layout, machining, assembling, and finishing.
Chapter 13 - Mechanical Engineering.pptx
Mechanical engineering focuses on the design, production, analysis, and improvement of technical objects. This chapter discusses key topics in mechanical engineering including linking components, guiding motion, different types of motion transmission systems, and motion transformation. Guiding components control the motion of moving parts and can ensure translational, rotational, or helical motion. Motion transmission systems relay motion from one part to another and include gear trains, chain and sprocket systems, belt and pulley systems, and others. Motion transformation changes the type of motion and is achieved through systems like rack and pinion gears, screw gears, cam and follower mechanisms, and sliders.
Physical & Chemical Changes
This document discusses physical and chemical changes. It defines physical changes as changes that do not alter the chemical composition of a substance and are typically reversible, such as freezing or dissolving. Chemical changes alter the chemical composition by producing new substances, are not easily reversible, and result in changes like formation of a gas or precipitate, changes in color or mass. Examples of each type of change are provided along with signs that indicate whether a change is physical or chemical. Common chemical reactions like synthesis, decomposition, precipitation, and acid-base neutralization are also outlined.
Compounds & Elements
This document defines key chemistry concepts including matter, mixtures, pure substances, elements, and compounds. It states that matter is anything that has mass and takes up space. Mixtures are combinations of two or more substances that are not chemically combined, while pure substances are either elements or compounds. Elements are the simplest forms of matter that cannot be broken down further, while compounds are formed by two or more elements that are chemically bonded together. The document provides examples of each term and guidelines for identifying elements and compounds based on changes in mass during chemical reactions.
Atomic Theory Overview
The document summarizes the historical development of atomic structure models from ancient philosophers to modern atomic theory. It describes early continuous models proposed by Aristotle and Democritus' earliest concept of atoms. John Dalton established the foundations of modern atomic theory by proposing atoms as indivisible particles that combine in whole number ratios. J.J. Thomson's experiments led him to propose a "plum pudding" model with electrons embedded in a positive sphere. Rutherford determined atoms have a small, dense positively charged nucleus surrounded by electrons. Modern atomic structure incorporates discoveries by Bohr and Chadwick that atoms consist of a nucleus of protons and neutrons surrounded by electrons in shells.
Moles
This document provides information about moles, molar mass, and stoichiometry calculations. It defines a mole as the amount of substance containing 6.02 x 1023 particles and molar mass as the mass of one mole of a substance. Formulas are given for calculating molar mass from elemental compositions and for converting between mass and moles using molar mass. Examples show calculations of molar mass for compounds and conversions between grams and moles for different substances.
Periodic Table
The document discusses key concepts about the periodic table including:
- Locating metals, nonmetals, and metalloids and identifying families such as alkali metals, alkaline earth metals, halogens, and inert gases.
- Describing properties of metals, nonmetals, and metalloids and how they differ.
- Explaining that periods indicate the number of electron shells an element has and families indicate the number of valence electrons.
- Identifying important families on the periodic table including alkali metals, alkaline earth metals, halogens, and inert gases.
Isotopes
Isotopes are atoms of the same element that have different numbers of neutrons. They have the same number of protons and electrons, so they have the same chemical properties. Some isotopes are stable while others are unstable and radioactive. Both stable and radioactive isotopes occur naturally, but radioisotopes are also produced artificially. Radioisotopes have important applications in medicine, industry, research and other areas due to their radioactive properties.
Atomic Structure Radioactivity
Rutherford's gold foil experiment led to a new atomic model. When alpha particles were fired at a thin gold foil, most passed through without deflection, showing atoms are mostly empty space. Some particles bounced straight back, indicating a small, dense nucleus. Others were slightly deflected, proving the nucleus has a positive charge and is the location of an atom's mass. This led Rutherford to propose an atomic model with a small, dense, positively charged nucleus surrounded by orbiting electrons.
Bohr Rutherford Atomic Model
The document describes the Bohr-Rutherford atomic model. It states that the atom consists of a small, dense nucleus containing protons and neutrons, surrounded by electrons in shells. The atomic number equals the number of protons, and a neutral atom has the same number of protons and electrons. Electrons are located in shells outside the nucleus, with the first shell holding 2 electrons and subsequent shells able to hold more. Diagrams illustrate atoms showing their arrangement of subatomic particles. Examples are provided of looking up atomic properties and drawing Bohr-Rutherford diagrams for different elements.
Thompson & Rutherford
This document discusses J.J. Thomson and Ernest Rutherford's atomic models. It summarizes Thomson's cathode ray experiments that discovered the electron. Thomson believed atoms were a uniform positive sphere with electrons embedded in it. The document also describes Rutherford's model with a small, dense positive nucleus surrounded by electrons in orbit. It lists the key properties of cathode rays that revealed electrons are negatively charged and have mass.
Atomic Structure
This document discusses early atomic theories and models of matter. It describes the theories of Aristotle, who believed that matter was continuous with no empty spaces, and Democritus, who proposed that matter was discontinuous and could be divided into smaller particles called atoms. The document then focuses on John Dalton's atomic model from the early 1800s, which proposed that all matter is composed of atoms, atoms of the same element are identical, atoms of different elements differ, and atoms combine in chemical reactions. Dalton's model helped establish the foundations of modern atomic theory.
Vectors Victor
- A vector is a quantity that has both magnitude and direction, represented by an arrow.
- The magnitude (length) of a vector AB is written as ||AB|| and can be calculated as √((x2-x1)2 + (y2-y1)2) for vectors in the Cartesian plane.
- Common vector operations include addition, subtraction, scalar multiplication, and determining the resultant. Chasles' Rule states that AB + BC = AC.
- Vectors can be equal, opposite, collinear, orthogonal, or the zero vector (with magnitude 0 and no direction).
Trig cheat sheet
The document defines trigonometric functions using right triangles and the unit circle. It lists properties of the trig functions including domain, range, period, formulas, and identities. It also covers inverse trig functions, the laws of sines, cosines, and tangents, and the unit circle.
Trig identities
This document discusses trigonometric identities and how to prove trigonometric statements. It covers basic trigonometric ratios and inverse ratios. It then presents three important trigonometric identities: 1) sin^2(θ) + cos^2(θ) = 1, 2) 1 + tan^2(θ) = sec^2(θ), and 3) 1 + cot^2(θ) = csc^2(θ). It provides examples of proving identities by simplifying expressions using basic identities, operations, and factoring. It also includes sample exam questions on proving an identity and determining an equivalent expression for tan(A) + cot(A).
Trig functions
This document covers trigonometric functions including ratios, identities, and tables of trig values. It introduces radians as another way to measure angles and compares radians to degrees. Formulas are provided for converting between radians and degrees as well as calculating arc length. Examples demonstrate using radians with trig functions and finding arc lengths. The document concludes with activities involving solving problems related to trig ratios, arc measures, and conversions between radians and degrees.
Parabola
The document discusses parabolas and their key properties. A parabola is defined as the locus of points equidistant from a focal point and directrix. The document provides the standard form equations of parabolas with their focal points located at various positions, and explains that transformed parabolas can be defined with shifts in the x and y variables. Examples are given of how to determine if a point lies inside or outside the parabola bounds. Two exam questions are presented where the correct answer is a parabola, due to the geometric description matching that of a parabola.
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Exponents and Logs
- 1. Chapter 4 Logarithms & Exponentials and Their Functions
- 2. Exponents Exponential mathematics involves working with a base term, c, that is multiplied by itself, m times, or the exponent is m. This equals the power, p, of a number. p = c m E.g. 9 = 3 2
- 3. Exponent Math Rules 1: Review Multiplying Exponent: c a x c b = c a + b (N.B. like bases) E.g. 4 2 x 4 3 = 4 2 + 3 = 4 5 Dividing Exponents: c a / c b = c a - b (N.B. like bases) E.g. 4 2 / 4 3 = 4 2 - 3 = 4 -1 = ¼ N.B. A negative exponent is a reciprocal under 1.
- 4. Math Exponent Rules 2: Review Exponent to an Exponent: (c a )b = cab (7 2 )3 = 7 2 x 72 x 72 = 7 2x3 = 7 6 The nth Root: c 1/n = n√c 25 ½ = √25 = 5 8 1/3 = 3 √8 = 2
- 5. Math Exponent Rules 3: Review Scientific Notation: a.bcd x 10 e 1.234 x 105 = 123400.(d.p.5 places) 1.23 x 10 -4 = 0.0123 (d.p. 4 places) Combining: a b x c d = a b x c d if bases are not same
- 6. Activities Go to page 347 Do Q. I, j Go to page 359 Do Q. 11, 14
- 7. Logarithms Rules 1: Fundamental Law: The logarithm is the inverse of an exponent. c log c m = m E.g. 3 log 3 5 = 5 This can be used to “get rid of” the log function. E.g. log 3 x = 4 3^ = 3^ x = 3 4 = 81
- 8. Logarithm Rules 2: Product Law: log c mn = log c m + log c n E.g. log 5 40 = log 5 8 + log 5 5 Quotient Law: log c m/n = log c m - log c n E.g. log 5 40 = log 5 8 - log 5 5
- 9. Logarithm Rules 3: Exponent of a logarithm: log c m n = n log c m E.g. log 7 11 2 = 2 log 7 11 Change of Base: Log c m = log s m/log s c Where c,m,s Є R+, c ≠ 1
- 10. Logarithm Rules 3: Log 3 5 = log 10 3 / log 10 5 Base is 10, so you can calculate this. = 0.477 / 0.699 = 0.683
- 11. Activity Go to page 397 And do Q. 1-3abc, 6a-d 7,13, 16, 19
- 12. Graph of Exponential Function
- 13. Properties of Basic & Transformed Rule y = c x y = ac x +k Graph Asymptote y = 0 (0,1) (1,c) Asymptote y=k Domain R R Range ]0,∞[ If a<0, ]-∞,k[ If a>0, ]k,∞[ Zero None if a/k= neg # Extremes None None Sign Positive Depends on zero Variation c>1 inc 0<c<1 dec Inc or dec Inverse Log fn. Log fn.
- 14. Graph of Logarithmic Function Red is to base e, green is to base 10, and purple is to base 1.7. Logs of all bases pass through the point (1, 0), because any number raised to the power 0 is 1, and through the points (b, 1) for base b, because a number raised to the power 1 is itself. The y axis or line x = 0 is an asymptote.
- 15. Properties of Basic & Transformed Rule y = logc x y=alogc(b(x-h))+k Graph Asymptote (c, 0),(1/c,-1) Asymptote x=h Domain ]0,∞[ If b<0, ]-∞,h[ If b>0, ]h,∞[ Range R R Zero 1 Only one Extremes None None Sign Depends Depends Variation c>1 inc 0<c<1 dec Depends on a&b sign Inverse Exp. Fn. Exp. Fn.
- 16. Activity Exponential graph questions Go to page 359; Do Q. 15, 20 Go to page 366; Do Q. 6, 13 Logarithmic graph questions Go to page 408; Do Q. 1a, 3 Go to page 411; Do Q. 18
- 17. Activity Go to page 424, Do Q. 1-8 Go to page 386, Do Q. 1-8 Log & Exp Website