This document provides an overview of indices and logarithms in elementary mathematics. It begins by defining integer indices and establishing laws for integer indices. It then extends the definition of indices to rational numbers by defining qth roots. Laws of indices are generalized to apply to rational exponents. Examples are provided to illustrate working with rational exponents. The chapter then introduces exponential functions, defining them as continuous functions that pass through the points (k, ak) for rational k. Laws for exponential functions are stated for real exponents.
This document contains a 3-page excerpt from the textbook "Elementary Mathematics" by W W L Chen and X T Duong. The excerpt discusses basic algebra concepts including:
- The real number system and subsets like natural numbers, integers, rational numbers, irrational numbers
- Rules of arithmetic operations like addition, subtraction, multiplication, division
- Properties of square roots
- Distributive laws for multiplication
- Arithmetic of fractions including addition and subtraction of fractions.
The document is a lesson on implicit differentiation and related concepts:
1) Implicit differentiation allows one to take the derivative of an implicitly defined relation between x and y, even if y is not explicitly defined as a function of x.
2) Examples are provided to demonstrate implicit differentiation, such as finding the slope of a tangent line to a curve.
3) The van der Waals equation is introduced to describe non-ideal gas properties, and implicit differentiation is used to find the isothermal compressibility of a van der Waals gas.
The document discusses partial differential equations and their solutions. It can be summarized as:
1) A partial differential equation involves a function of two or more variables and some of its partial derivatives, with one dependent variable and one or more independent variables. Standard notation is presented for partial derivatives.
2) Partial differential equations can be formed by eliminating arbitrary constants or arbitrary functions from an equation relating the dependent and independent variables. Examples of each method are provided.
3) Solutions to partial differential equations can be complete, containing the maximum number of arbitrary constants allowed, particular where the constants are given specific values, or singular where no constants are present. Methods for determining the general solution are described.
5 4 equations that may be reduced to quadraticsmath123b
The document discusses reducing equations to quadratic equations using substitution. It explains that if a pattern is repeated in an expression, a variable can be substituted to simplify the equation. Examples show substituting (x/(x-1)) with y, then solving the resulting quadratic equation for y and back substituting to find values of x. This reduction technique converts difficult equations into two easier equations to solve: the quadratic after substitution and then solving for the original variable.
1) The document discusses solving partial differential equations using variable separation. It presents the one-dimensional wave equation and solves it using variable separation.
2) It derives three cases for the solution based on whether k is positive, negative, or zero. It then presents the general solution as a summation involving sines and cosines.
3) It applies the general solution to two example problems of a vibrating string, finding the displacement as a function of position and time by satisfying the boundary conditions.
The document defines matrices and their properties, including symmetric, skew-symmetric, and determinant. It provides examples of solving systems of equations using matrices and their inverses. It also discusses properties of determinants, including properties related to symmetric and skew-symmetric matrices. Inverse trigonometric functions are defined, including their domains, ranges, and relationships between inverse functions using addition and subtraction formulas. Sample problems are provided to solve systems of equations and evaluate determinants.
5 4 equations that may be reduced to quadratics-xmath123b
The document discusses reducing equations to quadratic equations using substitution. It explains that if a pattern is repeated in an expression, a variable can be substituted to simplify the equation. Examples show substituting expressions like (x/(x-1)) with a variable y, solving the resulting quadratic equation for y, and then substituting back to find values of x. This process of solving two simpler equations through substitution is demonstrated to solve equations that are otherwise difficult to solve directly.
(1) The document discusses various integration techniques including: review of integral formulas, integration by parts, trigonometric integrals involving products of sines and cosines, trigonometric substitutions, and integration of rational functions using partial fractions.
(2) Examples are provided to demonstrate each technique, such as using integration by parts to evaluate integrals of the form ∫udv, using trigonometric identities to reduce powers of trigonometric functions, and using partial fractions to break down rational functions into simpler fractions.
(3) The key techniques discussed are integration by parts, trigonometric substitutions to transform integrals involving quadratic expressions into simpler forms, and partial fractions to decompose rational functions for integration. Various examples illustrate the
This document contains a 3-page excerpt from the textbook "Elementary Mathematics" by W W L Chen and X T Duong. The excerpt discusses basic algebra concepts including:
- The real number system and subsets like natural numbers, integers, rational numbers, irrational numbers
- Rules of arithmetic operations like addition, subtraction, multiplication, division
- Properties of square roots
- Distributive laws for multiplication
- Arithmetic of fractions including addition and subtraction of fractions.
The document is a lesson on implicit differentiation and related concepts:
1) Implicit differentiation allows one to take the derivative of an implicitly defined relation between x and y, even if y is not explicitly defined as a function of x.
2) Examples are provided to demonstrate implicit differentiation, such as finding the slope of a tangent line to a curve.
3) The van der Waals equation is introduced to describe non-ideal gas properties, and implicit differentiation is used to find the isothermal compressibility of a van der Waals gas.
The document discusses partial differential equations and their solutions. It can be summarized as:
1) A partial differential equation involves a function of two or more variables and some of its partial derivatives, with one dependent variable and one or more independent variables. Standard notation is presented for partial derivatives.
2) Partial differential equations can be formed by eliminating arbitrary constants or arbitrary functions from an equation relating the dependent and independent variables. Examples of each method are provided.
3) Solutions to partial differential equations can be complete, containing the maximum number of arbitrary constants allowed, particular where the constants are given specific values, or singular where no constants are present. Methods for determining the general solution are described.
5 4 equations that may be reduced to quadraticsmath123b
The document discusses reducing equations to quadratic equations using substitution. It explains that if a pattern is repeated in an expression, a variable can be substituted to simplify the equation. Examples show substituting (x/(x-1)) with y, then solving the resulting quadratic equation for y and back substituting to find values of x. This reduction technique converts difficult equations into two easier equations to solve: the quadratic after substitution and then solving for the original variable.
1) The document discusses solving partial differential equations using variable separation. It presents the one-dimensional wave equation and solves it using variable separation.
2) It derives three cases for the solution based on whether k is positive, negative, or zero. It then presents the general solution as a summation involving sines and cosines.
3) It applies the general solution to two example problems of a vibrating string, finding the displacement as a function of position and time by satisfying the boundary conditions.
The document defines matrices and their properties, including symmetric, skew-symmetric, and determinant. It provides examples of solving systems of equations using matrices and their inverses. It also discusses properties of determinants, including properties related to symmetric and skew-symmetric matrices. Inverse trigonometric functions are defined, including their domains, ranges, and relationships between inverse functions using addition and subtraction formulas. Sample problems are provided to solve systems of equations and evaluate determinants.
5 4 equations that may be reduced to quadratics-xmath123b
The document discusses reducing equations to quadratic equations using substitution. It explains that if a pattern is repeated in an expression, a variable can be substituted to simplify the equation. Examples show substituting expressions like (x/(x-1)) with a variable y, solving the resulting quadratic equation for y, and then substituting back to find values of x. This process of solving two simpler equations through substitution is demonstrated to solve equations that are otherwise difficult to solve directly.
(1) The document discusses various integration techniques including: review of integral formulas, integration by parts, trigonometric integrals involving products of sines and cosines, trigonometric substitutions, and integration of rational functions using partial fractions.
(2) Examples are provided to demonstrate each technique, such as using integration by parts to evaluate integrals of the form ∫udv, using trigonometric identities to reduce powers of trigonometric functions, and using partial fractions to break down rational functions into simpler fractions.
(3) The key techniques discussed are integration by parts, trigonometric substitutions to transform integrals involving quadratic expressions into simpler forms, and partial fractions to decompose rational functions for integration. Various examples illustrate the
This document contains mathematical formula tables including:
1. Greek alphabet, indices and logarithms, trigonometric identities, complex numbers, hyperbolic identities, and series.
2. Derivatives of common functions, product rule, quotient rule, chain rule, and Leibnitz's theorem.
3. Integrals of common functions, double integrals, and the substitution rule for integrals.
The document discusses solving the two-dimensional Laplace equation to model steady heat flow problems. It presents:
1) The general boundary value problem (BVP) for the Laplace equation in a semi-infinite and finite lamina.
2) The separation of variables method to obtain solutions as a sum of products of ordinary differential equations.
3) Applying boundary conditions to determine constants and obtain the general solution for temperature distribution.
4) Examples of applying the method to specific BVPs for steady heat flow, including plates with various boundary temperature profiles and geometries.
The document discusses solving systems of nonlinear equations in two variables. It provides examples of nonlinear systems that contain equations that are not in the form Ax + By = C, such as x^2 = 2y + 10. Methods for solving nonlinear systems include substitution and addition. The substitution method involves solving one equation for one variable and substituting into the other equation. The addition method involves rewriting the equations and adding them to eliminate variables. Examples demonstrate both methods and finding the solution set that satisfies both equations.
The document discusses how to solve radical equations by squaring both sides of the equation repeatedly to remove radicals. Key steps include:
1) Isolating the radical term to one side of the equation before squaring.
2) Using the identity (a ± b)2 = a2 ± 2ab + b2 to expand squared terms.
3) Squaring both sides and solving the resulting non-radical equation for the variable.
4) Checking that solutions satisfy the original radical equation. Examples demonstrate these techniques.
Conformable Chebyshev differential equation of first kindIJECEIAES
In this paper, the Chebyshev-I conformable differential equation is considered. A proper power series is examined; there are two solutions, the even solution and the odd solution. The Rodrigues’ type formula is also allocated for the conformable Chebyshev-I polynomials.
Third-kind Chebyshev Polynomials Vr(x) in Collocation Methods of Solving Boun...IOSR Journals
This paper proposed the use of third-kind Chebyshev polynomials as trial functions in solving boundary value problems via collocation method. In applying this method, two different collocation points are considered, which are points at zeros of third-kind Chebyshev polynomials and equally-spaced points. These points yielded different results on each considered problem, thus possessing different level of accuracy. The
method is computational very simple and attractive. Applications are equally demonstrated through numerical examples to illustrate the efficiency and simplicity of the approach
Linearprog, Reading Materials for Operational Research Derbew Tesfa
The document discusses linear programming (LP), which involves optimizing a linear objective function subject to linear constraints. It provides examples of LP problems, such as production planning and transportation problems. It defines key LP concepts like the feasible region, basic solutions, basic variables, and degenerate basic feasible solutions. It also describes how to transform any LP problem into standard form and discusses properties of optimal solutions.
This document contains an introduction and table of contents for a chapter on differentiation of functions with several variables. It covers topics like limits and continuity, partial derivatives, chain rules, directional derivatives, gradients, tangent planes, and finding extrema. The chapter introduces concepts like functions of two or more variables, limits, continuity, partial derivatives and their properties, implicit differentiation using the chain rule, and finding directional derivatives and gradients. It provides definitions, theorems, examples and illustrations of key concepts in multivariable calculus.
This document contains solutions to 5 problems posed at the IMC 2017 conference. The solutions are summarized as follows:
1) The possible eigenvalues of the matrix A described in Problem 1 are 0, 1, and -1±√3i/2.
2) Problem 2 proves that for any differentiable function f satisfying the Lipschitz condition, f(x)2 < 2Lf(x) for all x.
3) Problem 3 shows that for any set S subset of {1,2,...,2017}, there exists an integer n such that the sequence ak(n) defined in the problem satisfies the property that ak(n) is a perfect square if and only if k is
Let Pn(x) be the Legendre polynomial of degree n. Then the generating function for Pn(x) is given by:
∞
1
Pn(x)tn = √
n=0
1 − 2xt + t2
Differentiating both sides with respect to t, we get:
∞
∑nPn(x)tn-1 = -xt(1 − 2xt + t2)-1/2 + (1 − 2xt + t2)-3/2
n=1
Multiplying both sides by (1 − 2xt + t2)1/2, we get:
∞
∑
Lesson 29: Integration by Substition (worksheet solutions)Matthew Leingang
The document provides solutions to 11 integrals using substitution methods. The integrals involve trigonometric, exponential, logarithmic and algebraic functions. Substitutions are given for some integrals and need to be determined for others. The solutions show setting up the substitution, finding the differential, integrating and solving for the antiderivative. Some integrals have multiple valid substitution options. One integral is solved using a trigonometric identity in addition to substitution.
The document describes three methods for solving second degree equations (ax2 + bx + c = 0):
1) The square-root method, which is used when the x-term is missing. It involves solving for x2 and taking the square root to find x.
2) Factoring, which involves factoring the equation into the form (ax + b)(cx + d) = 0. It can only be used if b2 - 4ac is a perfect square.
3) The quadratic formula, which can be used to solve any second degree equation.
The document provides examples and explanations for solving different types of equations, including:
1) Polynomial equations through factoring or the quadratic formula.
2) Rational equations by clearing denominators.
3) Radical equations by squaring both sides to remove radicals.
4) Absolute value equations by recognizing that |x-c|=r implies x=c±r.
The document also discusses solving power equations, finding zeros and domains of functions, and using properties of absolute values.
The document discusses solving linear equations using numerical methods in MATLAB. It provides code implementations for Gauss-Seidel method, Jacobi method, and Cholesky method. It also includes routines for calculating the inverse of a 3x3 matrix, matrix norm, and forward substitution. Examples are given applying each method to different 3x3 matrices. Spectral radius is also calculated for some examples to analyze convergence properties.
This document discusses methods for solving numerical equations, including the bisection method, Newton-Raphson method, and method of false position. It provides definitions and step-by-step computations for each method. For the bisection method, it gives an example of finding the positive root of x3 - x = 1. For Newton-Raphson, it gives examples of finding the root of 2x3 - 3x - 6 = 0 and x3 = 6x - 4. The document serves to introduce numerical methods for solving equations.
Quadratic equations take the form ax^2 + bx + c = 0. This document discusses four methods for solving quadratic equations: factorizing, completing the square, using the quadratic formula, and graphing. It provides examples of solving quadratic equations with each method and emphasizes that practice is needed to master the techniques.
The document discusses the algebra of radicals. It provides the multiplication rule that √xy = √x√y and √xx = x. It also provides the division rule. It then gives examples of simplifying radical expressions using these rules, such as √3 * √3 = 3, 3√3 * √3 = 9, and (3√3)2 = 27.
This document defines continuity and uniform continuity of functions. A function f is continuous on a set S if small changes in the input x result in small changes in the output f(x). A function is uniformly continuous if the same relationship holds for all inputs and outputs simultaneously, not just for a fixed input. Several examples are provided to illustrate the difference. The key difference is that a continuous function may depend on the specific input point, while a uniformly continuous function does not. Functions that satisfy a Lipschitz inequality are proven to be uniformly continuous.
The document describes the Jacobi iterative method for solving systems of linear equations. It explains that the Jacobi method makes approximations to the solution by iteratively solving for each variable in terms of the most recent approximations for the other variables. The method works by rewriting the system of equations in a form where the coefficient matrix is split into diagonal, lower triangular, and upper triangular matrices. Approximations converge to the true solution as the number of iterations increase. Pseudocode and a MATLAB implementation are provided.
Radical equations are equations with an unknown variable under a radical sign. To solve radical equations, each side of the equation is squared repeatedly to remove all radicals. This is done because if two expressions are equal, then their squares are also equal. Once all radicals are removed, the resulting equation can be solved normally for the unknown variable. Examples show how to isolate radical terms, expand squared expressions using formulas, and check solutions. Squaring each side must be done carefully to properly isolate radical terms.
This document discusses different methods for allocating joint costs among joint products. It begins by defining key terms like joint costs, joint products, byproducts, and splitoff point. It then explains four different approaches to allocating joint costs: 1) market-based methods including sales value at splitoff, estimated net realizable value, and constant gross margin percentage; and 2) physical measure methods. Examples are provided to illustrate how to allocate joint costs using the sales value at splitoff and estimated net realizable value methods.
The document provides teaching materials about the digestive system for grades 9-12, including standards, related links, discussion questions, activities, and reproducible materials. It aims to educate students about how their digestive systems work to power their bodies and the importance of making good nutrition and exercise decisions. Key points covered include the roles of different organs like the stomach, small intestine, and liver in breaking down food and absorbing nutrients into the bloodstream. Common minor digestive issues are also discussed.
This document contains mathematical formula tables including:
1. Greek alphabet, indices and logarithms, trigonometric identities, complex numbers, hyperbolic identities, and series.
2. Derivatives of common functions, product rule, quotient rule, chain rule, and Leibnitz's theorem.
3. Integrals of common functions, double integrals, and the substitution rule for integrals.
The document discusses solving the two-dimensional Laplace equation to model steady heat flow problems. It presents:
1) The general boundary value problem (BVP) for the Laplace equation in a semi-infinite and finite lamina.
2) The separation of variables method to obtain solutions as a sum of products of ordinary differential equations.
3) Applying boundary conditions to determine constants and obtain the general solution for temperature distribution.
4) Examples of applying the method to specific BVPs for steady heat flow, including plates with various boundary temperature profiles and geometries.
The document discusses solving systems of nonlinear equations in two variables. It provides examples of nonlinear systems that contain equations that are not in the form Ax + By = C, such as x^2 = 2y + 10. Methods for solving nonlinear systems include substitution and addition. The substitution method involves solving one equation for one variable and substituting into the other equation. The addition method involves rewriting the equations and adding them to eliminate variables. Examples demonstrate both methods and finding the solution set that satisfies both equations.
The document discusses how to solve radical equations by squaring both sides of the equation repeatedly to remove radicals. Key steps include:
1) Isolating the radical term to one side of the equation before squaring.
2) Using the identity (a ± b)2 = a2 ± 2ab + b2 to expand squared terms.
3) Squaring both sides and solving the resulting non-radical equation for the variable.
4) Checking that solutions satisfy the original radical equation. Examples demonstrate these techniques.
Conformable Chebyshev differential equation of first kindIJECEIAES
In this paper, the Chebyshev-I conformable differential equation is considered. A proper power series is examined; there are two solutions, the even solution and the odd solution. The Rodrigues’ type formula is also allocated for the conformable Chebyshev-I polynomials.
Third-kind Chebyshev Polynomials Vr(x) in Collocation Methods of Solving Boun...IOSR Journals
This paper proposed the use of third-kind Chebyshev polynomials as trial functions in solving boundary value problems via collocation method. In applying this method, two different collocation points are considered, which are points at zeros of third-kind Chebyshev polynomials and equally-spaced points. These points yielded different results on each considered problem, thus possessing different level of accuracy. The
method is computational very simple and attractive. Applications are equally demonstrated through numerical examples to illustrate the efficiency and simplicity of the approach
Linearprog, Reading Materials for Operational Research Derbew Tesfa
The document discusses linear programming (LP), which involves optimizing a linear objective function subject to linear constraints. It provides examples of LP problems, such as production planning and transportation problems. It defines key LP concepts like the feasible region, basic solutions, basic variables, and degenerate basic feasible solutions. It also describes how to transform any LP problem into standard form and discusses properties of optimal solutions.
This document contains an introduction and table of contents for a chapter on differentiation of functions with several variables. It covers topics like limits and continuity, partial derivatives, chain rules, directional derivatives, gradients, tangent planes, and finding extrema. The chapter introduces concepts like functions of two or more variables, limits, continuity, partial derivatives and their properties, implicit differentiation using the chain rule, and finding directional derivatives and gradients. It provides definitions, theorems, examples and illustrations of key concepts in multivariable calculus.
This document contains solutions to 5 problems posed at the IMC 2017 conference. The solutions are summarized as follows:
1) The possible eigenvalues of the matrix A described in Problem 1 are 0, 1, and -1±√3i/2.
2) Problem 2 proves that for any differentiable function f satisfying the Lipschitz condition, f(x)2 < 2Lf(x) for all x.
3) Problem 3 shows that for any set S subset of {1,2,...,2017}, there exists an integer n such that the sequence ak(n) defined in the problem satisfies the property that ak(n) is a perfect square if and only if k is
Let Pn(x) be the Legendre polynomial of degree n. Then the generating function for Pn(x) is given by:
∞
1
Pn(x)tn = √
n=0
1 − 2xt + t2
Differentiating both sides with respect to t, we get:
∞
∑nPn(x)tn-1 = -xt(1 − 2xt + t2)-1/2 + (1 − 2xt + t2)-3/2
n=1
Multiplying both sides by (1 − 2xt + t2)1/2, we get:
∞
∑
Lesson 29: Integration by Substition (worksheet solutions)Matthew Leingang
The document provides solutions to 11 integrals using substitution methods. The integrals involve trigonometric, exponential, logarithmic and algebraic functions. Substitutions are given for some integrals and need to be determined for others. The solutions show setting up the substitution, finding the differential, integrating and solving for the antiderivative. Some integrals have multiple valid substitution options. One integral is solved using a trigonometric identity in addition to substitution.
The document describes three methods for solving second degree equations (ax2 + bx + c = 0):
1) The square-root method, which is used when the x-term is missing. It involves solving for x2 and taking the square root to find x.
2) Factoring, which involves factoring the equation into the form (ax + b)(cx + d) = 0. It can only be used if b2 - 4ac is a perfect square.
3) The quadratic formula, which can be used to solve any second degree equation.
The document provides examples and explanations for solving different types of equations, including:
1) Polynomial equations through factoring or the quadratic formula.
2) Rational equations by clearing denominators.
3) Radical equations by squaring both sides to remove radicals.
4) Absolute value equations by recognizing that |x-c|=r implies x=c±r.
The document also discusses solving power equations, finding zeros and domains of functions, and using properties of absolute values.
The document discusses solving linear equations using numerical methods in MATLAB. It provides code implementations for Gauss-Seidel method, Jacobi method, and Cholesky method. It also includes routines for calculating the inverse of a 3x3 matrix, matrix norm, and forward substitution. Examples are given applying each method to different 3x3 matrices. Spectral radius is also calculated for some examples to analyze convergence properties.
This document discusses methods for solving numerical equations, including the bisection method, Newton-Raphson method, and method of false position. It provides definitions and step-by-step computations for each method. For the bisection method, it gives an example of finding the positive root of x3 - x = 1. For Newton-Raphson, it gives examples of finding the root of 2x3 - 3x - 6 = 0 and x3 = 6x - 4. The document serves to introduce numerical methods for solving equations.
Quadratic equations take the form ax^2 + bx + c = 0. This document discusses four methods for solving quadratic equations: factorizing, completing the square, using the quadratic formula, and graphing. It provides examples of solving quadratic equations with each method and emphasizes that practice is needed to master the techniques.
The document discusses the algebra of radicals. It provides the multiplication rule that √xy = √x√y and √xx = x. It also provides the division rule. It then gives examples of simplifying radical expressions using these rules, such as √3 * √3 = 3, 3√3 * √3 = 9, and (3√3)2 = 27.
This document defines continuity and uniform continuity of functions. A function f is continuous on a set S if small changes in the input x result in small changes in the output f(x). A function is uniformly continuous if the same relationship holds for all inputs and outputs simultaneously, not just for a fixed input. Several examples are provided to illustrate the difference. The key difference is that a continuous function may depend on the specific input point, while a uniformly continuous function does not. Functions that satisfy a Lipschitz inequality are proven to be uniformly continuous.
The document describes the Jacobi iterative method for solving systems of linear equations. It explains that the Jacobi method makes approximations to the solution by iteratively solving for each variable in terms of the most recent approximations for the other variables. The method works by rewriting the system of equations in a form where the coefficient matrix is split into diagonal, lower triangular, and upper triangular matrices. Approximations converge to the true solution as the number of iterations increase. Pseudocode and a MATLAB implementation are provided.
Radical equations are equations with an unknown variable under a radical sign. To solve radical equations, each side of the equation is squared repeatedly to remove all radicals. This is done because if two expressions are equal, then their squares are also equal. Once all radicals are removed, the resulting equation can be solved normally for the unknown variable. Examples show how to isolate radical terms, expand squared expressions using formulas, and check solutions. Squaring each side must be done carefully to properly isolate radical terms.
This document discusses different methods for allocating joint costs among joint products. It begins by defining key terms like joint costs, joint products, byproducts, and splitoff point. It then explains four different approaches to allocating joint costs: 1) market-based methods including sales value at splitoff, estimated net realizable value, and constant gross margin percentage; and 2) physical measure methods. Examples are provided to illustrate how to allocate joint costs using the sales value at splitoff and estimated net realizable value methods.
The document provides teaching materials about the digestive system for grades 9-12, including standards, related links, discussion questions, activities, and reproducible materials. It aims to educate students about how their digestive systems work to power their bodies and the importance of making good nutrition and exercise decisions. Key points covered include the roles of different organs like the stomach, small intestine, and liver in breaking down food and absorbing nutrients into the bloodstream. Common minor digestive issues are also discussed.
The document provides examples and explanations for solving various types of polynomial equations, beginning with linear equations and then simultaneous linear equations. It then introduces quadratic equations.
Key steps for solving linear equations include isolating the variable, usually by subtracting or dividing both sides by a constant. For simultaneous linear equations, common techniques are elimination (subtracting equations to remove a variable) and substitution.
Several examples demonstrate solving systems of two and three linear equations graphically or algebraically. The document emphasizes setting up equivalent equations to efficiently eliminate variables. Finally, quadratic equations are introduced but no examples are provided.
1. This document provides examples and definitions related to arithmetic and geometric progressions. It defines arithmetic and geometric progressions and gives formulas for finding individual terms and summing the total terms.
2. Key examples show how to pair terms in an arithmetic progression to simplify summing the total. It also illustrates summing fractions in a geometric progression as areas in a rectangle.
3. Formulas are provided for finding the k-th term and summing the total terms of an arithmetic or geometric progression, depending on whether the difference or ratio is constant between terms.
This chapter introduces matrices and their basic arithmetic operations. Matrices allow linear equations to be written in a compact matrix form. The key operations covered are:
1) Matrix addition and subtraction are performed element-wise.
2) A matrix can be multiplied by a scalar by multiplying each element.
3) Matrix multiplication involves multiplying the rows of the first matrix by the columns of the second.
4) For matrix multiplication to have an inverse, the matrices must satisfy a condition on their elements.
Cvd definitions and statistics jan 2012nahomyitbarek
This document discusses cardiovascular disease (CVD) epidemiology and concepts. It defines key terms like coronary artery disease, cardiovascular disease, and surrogate measures. It provides statistics on CVD prevalence, mortality trends, and costs in the United States. Graphs show trends in CVD procedures, prevalence by age and sex, deaths by disease, and more. Risk factors discussed include age, smoking, cholesterol levels, blood pressure, diabetes, and family history. The document also discusses lifetime and 10-year CVD risk according to the Framingham Heart Study.
This document is a laboratory manual for a microbiology course at the University of Lethbridge. It provides details on the course schedule, grading breakdown, safety guidelines, and 10 laboratory exercises covering topics like microscopy, bacterial morphology, nitrogen fixation, the Ames test, biochemical tests, yeast fermentation and virology. Appendices provide additional instructions on microscope use, aseptic technique, media preparation and laboratory notebook expectations. The manual aims to guide students through hands-on learning of key microbiology principles and techniques over the course of the semester.
This document discusses linear and quadratic functions. It begins by defining linear functions as having a constant slope and providing examples of linear relationships involving student grades and beer demand. It then discusses inverse linear functions and using linear functions to model tax rates. The document next discusses quadratic functions as having a u-shaped or hill-shaped graph depending on the coefficient of the x^2 term. It provides an example of solving a quadratic equation graphically and discusses how a quadratic equation can have 0, 1, or 2 solutions. The summary concludes by noting a special case where a quadratic function reduces to y=x^2.
1. Glucose is broken down into pyruvate through glycolysis, which yields 2 ATP and 2 NADH per glucose molecule.
2. In the mitochondria, pyruvate is broken down into acetyl CoA, producing CO2 and NADH. Acetyl CoA then enters the Krebs cycle.
3. The Krebs cycle further oxidizes acetyl CoA, producing CO2 and loading electron carriers NADH and FADH2. One ATP is also generated per acetyl CoA in the Krebs cycle. The electron carriers are used to power ATP production through oxidative phosphorylation.
This document summarizes an elementary mathematics chapter that introduces fundamental counting techniques:
1) It explains the fundamental principle of counting - that if event 1 can occur in n1 ways and event 2 in n2 ways, then the total number of ways for the events to occur together is n1 × n2.
2) It defines permutation as the number of arrangements of k objects from n objects, which is equal to n!/(n-k)!. Examples show counting license plate and word arrangements.
3) It defines combination as the number of choices of k unordered objects from n objects, which is equal to n!/k!(n-k)!. Examples show counting letter and exam question
The document provides information about a biology lesson on cells. It includes objectives, agendas, and instructions for assignments on organelles and their functions. Students are tasked with explaining how at least 5 organelles work together to make and use proteins. They will view cells under a microscope and identify various cell components by making wet mount slides from different organisms. The document outlines safety protocols for the lab and requirements for completing the assignments.
1) The document discusses EDTA (ethylenediaminetetraacetic acid) titrations for determining metal concentrations. EDTA can form strong complexes with most metal ions and is used as an analytical tool.
2) Key concepts covered include the chelate effect whereby multidentate ligands like EDTA form very strong complexes with metal ions. The document also discusses factors that affect EDTA titrations like pH and conditional formation constants.
3) Methods of EDTA titrations described briefly include direct titrations, back titrations, displacement titrations, indirect titrations and the use of masking agents.
The document outlines a class on cellular respiration, with objectives to write the chemical equation for cell respiration and identify its reactants and products. The class agenda includes notes on cell respiration, reviewing how the body converts energy through ATP production in mitochondria, and a discussion of the chemical formula for cell respiration.
This document outlines the biology syllabus for grades 9 and 10 in Ethiopia. It covers 6 units for grade 9 and 5 units for grade 10. The syllabus was revised based on a needs assessment to address issues like content overload, difficulty, and relevance. Key changes included simplifying content, integrating subjects like technology and agriculture, and focusing on competencies in knowledge, skills, and values. The approach is based on constructivism, emphasizing that learners actively acquire and construct their own understanding through social and language-based learning activities.
Concept presentation on chemical bonding (iris lo)nahomyitbarek
This document outlines a 4-day lesson plan for teaching ionic and covalent bonding to grade 11 chemistry students according to Ontario curriculum expectations. The lesson plan includes an overview of key concepts, common student misconceptions, and activities to address them. Day 1 involves classifying compounds and demonstrations. Day 2 introduces bonding concepts. Day 3 focuses on ionic and covalent bonding through models and videos. Day 4 examines polar covalent bonding through a demonstration and building molecular models. Assessment strategies are provided for each day.
The main difference between an interpreter and a compiler is that a compiler translates the entire program into machine code all at once, whereas an interpreter translates and executes code line-by-line. Compiled programs typically run faster since the machine code has been optimized, while interpreted programs identify errors faster since they are translated and executed incrementally. Compiled programs produce standalone executable files while interpreted programs require the interpreter each time they are run.
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1. ELEMENTARY MATHEMATICS
W W L CHEN and X T DUONG
c W W L Chen, X T Duong and Macquarie University, 1999.
This work is available free, in the hope that it will be useful.
Any part of this work may be reproduced or transmitted in any form or by any means, electronic or mechanical, including
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Chapter 4
INDICES AND LOGARITHMS
4.1. Indices
Given any non-zero real number a ∈ R and any natural number k ∈ N, we often write
ak = a × . . . × a . (1)
k
This definition can be extended to all integers k ∈ Z by writing
a0 = 1 (2)
and
1 1
ak = = (3)
a−k a × ... × a
−k
whenever k is a negative integer, noting that −k ∈ N in this case.
It is not too difficult to establish the following.
LAWS OF INTEGER INDICES. Suppose that a, b ∈ R are non-zero. Then for every m, n ∈ Z,
we have
(a) am an = am+n ;
am
(b) n = am−n ;
a
(c) (am )n = amn ; and
(d) (ab)m = am bm .
† This chapter was written at Macquarie University in 1999.
2. 4–2 W W L Chen and X T Duong : Elementary Mathematics
We now further extend the definition of ak to all rational numbers k ∈ Q. To do this, we first of all
need to discuss the q-th roots of a positive real number a, where q ∈ N. This is an extension of the idea
of square roots discussed in Section 1.2. We recall the following definition, slightly modified here.
Definition. Suppose that a ∈ R is positive. We say that x > 0 is the positive square root of a if
√
x2 = a. In this case, we write x = a.
We now make the following natural extension.
Definition. Suppose that a ∈ R is positive and q ∈ N. We say that x > 0 is the positive q-th root of
√
a if xq = a. In this case, we write x = q a = a1/q .
Recall now that every rational number k ∈ Q can be written in the form k = p/q, where p ∈ Z and
q ∈ N. We may, if we wish, assume that p/q is in lowest terms, where p and q have no common factors.
For any positive real number a ∈ R, we can now define ak by writing
ak = ap/q = (a1/q )p = (ap )1/q . (4)
In other words, we first of all calculate the positive q-th root of a, and then take the p-th power of this
q-th root. Alternatively, we can first of all take the p-th power of a, and then calculate the positive q-th
root of this p-th power.
We can establish the following generalization of the Laws of integer indices.
LAWS OF INDICES. Suppose that a, b ∈ R are positive. Then for every m, n ∈ Q, we have
(a) am an = am+n ;
am
(b) n = am−n ;
a
(c) (am )n = amn ; and
(d) (ab)m = am bm .
Remarks. (1) Note that we have to make the restriction that the real numbers a and b are positive.
If a = 0, then ak is clearly not defined when k is a negative integer. If a < 0, then we will have problems
taking square roots.
(2) It is possible to define cube roots of a negative real number a. It is a real number x satisfying
the requirement x3 = a. Note that x < 0 in this case. A similar argument applies to q-th roots when
q ∈ N is odd. However, if q ∈ N is even, then xq ≥ 0 for every x ∈ R, and so xq = a for any negative
a ∈ R. Hence a negative real number does not have real q-th roots for any even q ∈ N.
Example 4.1.1. We have 24 × 23 = 16 × 8 = 128 = 27 = 24+3 and 2−3 = 1/8.
Example 4.1.2. We have 82/3 = (81/3 )2 = 22 = 4. Alternatively, we have 82/3 = (82 )1/3 = 641/3 = 4.
√ √ √
Example 4.1.3. To show that 5 + 2 6 = 2+ 3, we first of all observe that both sides are positive,
and so it suffices to show that the squares of the two sides are equal. Note now that
√ √ √ √ √ √ √
( 2 + 3)2 = ( 2)2 + 2 2 3 + ( 3)2 = 5 + 2 × 21/2 31/2 = 5 + 2(2 × 3)1/2 = 5 + 2 6,
the square of the left hand side.
√ √ √
Example 4.1.4. To show that 8 − 4 3 = 6 − 2, it suffices to show that the squares of the two
sides are equal. Note now that
√ √ √ √ √ √
( 6 − 2)2 = ( 6)2 − 2 6 2 + ( 2)2 = 8 − 2 × 61/2 21/2 = 8 − 2(6 × 2)1/2
√ √ √
= 8 − 2 12 = 8 − 2 × 2 3 = 8 − 4 3,
the square of the left hand side.
3. Chapter 4 : Indices and Logarithms 4–3
In the remaining examples in this section, the variables x and y both represent positive real numbers.
Example 4.1.5. We have
2 2
x−1 × 2x1/2 = 2x(1/2)−1 = 2x−1/2 = =√ .
x1/2 x
Example 4.1.6. We have
1 1 x5/2 3
(27x2 )1/3 ÷ (x5 )1/2 = 271/3 (x2 )1/3 ÷ x5×(1/2) = 3x2/3 ÷ = 3x2/3 × 5/2
3 3 3 x
9
= 3x2/3 × 3x−5/2 = 9x2/3 x−5/2 = 9x(2/3)−(5/2) = 9x−11/6 = .
x11/6
Example 4.1.7. We have
(9x)1/2 (8x−1/2 )1/3 = 91/2 x1/2 × 81/3 x(−1/2)×(1/3) = 3x1/2 × 2x−1/6 = 6x(1/2)−(1/6) = 6x1/3 .
Example 4.1.8. We have
2 −2 −2
1 −1 1 −1 1
(8x3/4 )−2 ÷ x = (8x3/4 )−2 × x = 8−2 x(3/4)×(−2) × x(−1)×(−2)
2 2 2
√
1 −3/2 1 (−3/2)+2 1 1/2 x
= x × 22 x2 = x = x = .
82 16 16 16
Example 4.1.9. We have
√
8a2 b × a1/3 b5/3 = (8a2 b)1/3 a1/3 b5/3 = 81/3 (a2 )1/3 b1/3 a1/3 b5/3 = 2a2/3 b1/3 a1/3 b5/3
3
= 2a(2/3)+(1/3) b(1/3)+(5/3) = 2ab2 .
Example 4.1.10. We have
4
(16x1/6 y 2 )3 = ((16x1/6 y 2 )3 )1/4 = (16x1/6 y 2 )3/4 = 163/4 (x1/6 )3/4 (y 2 )3/4
= (161/4 )3 x(1/6)×(3/4) y 2×(3/4) = 8x1/8 y 3/2 .
We now miss out some intermediate steps in the examples below. The reader is advised to fill in all
the details in each step.
Example 4.1.11. We have
16(x2 y 3 )1/2 (4x6 y 4 )1/2 16xy 3/2 2x3 y 2 16xy 3/2 × 2x3 y 2 32x4 y 7/2
× = 3/2 3 × = 3/2 3 =
(2x1/2 y)3 (6x3 y 1/2 )2 8x y 36x6 y 8x y × 36x6 y 288x15/2 y 4
32 4 7/2 −15/2 −4 1 −7/2 −1/2 1
= x y x y = x y = 7/2 1/2 .
288 9 9x y
Example 4.1.12. We have
(3x)3 y 2 (5xy)4 (3x)3 y 2 (27x9 )3 (3x)3 y 2 × (27x9 )3 33 273 x30 y 2
2
÷ 9 )3
= 2
× 4
= 2 × (5xy)4
=
(5xy) (27x (5xy) (5xy) (5xy) 56 x6 y 6
531441 30 2 −6 −6 531441 24 −4 531441x24
= x y x y = x y = .
15625 15625 15625y 4
4. 4–4 W W L Chen and X T Duong : Elementary Mathematics
Example 4.1.13. We have
x−1 + y −1 x−1 − y −1 (x − y)(x−1 + y −1 ) − (x−1 − y −1 )(x + y)
− =
x+y x−y (x − y)(x + y)
(1 + xy −1 − yx−1 − 1) − (1 + yx−1 − xy −1 − 1) 2(xy −1 − yx−1 )
= =
(x − y)(x + y) (x − y)(x + y)
2 x y 2 x2 − y 2 2
= × − = × = .
(x − y)(x + y) y x (x − y)(x + y) xy xy
Example 4.1.14. We have
x−1 − y −1 1 1 1 1 y − x y 2 − x2
−2 − y −2
= (x−1 − y −1 ) ÷ (x−2 − y −2 ) = − ÷ 2
− 2 = ÷
x x y x y xy x2 y 2
y−x x2 y 2 x2 y 2 (y − x) x2 y 2 (y − x) xy
= × 2 = = = .
xy y −x 2 xy(y 2 − x2 ) xy(y − x)(y + x) x+y
Alternatively, write a = x−1 and b = y −1 . Then
−1 −1
x−1 − y −1 a−b 1 1 1 x+y xy
−2 − y −2
= 2 = = (a + b)−1 = + = = .
x a −b 2 a+b x y xy x+y
Example 4.1.15. We have
x2 1 1 x2 x+y x2 xy x3
x2 y −1 ÷ (x−1 + y −1 ) = ÷ + = ÷ = × = .
y x y y xy y x+y x+y
Example 4.1.16. We have
−1
1
xy ÷ ((x−1 + y)−1 )−1 = xy × (x−1 + y)−1 = xy × +y
x
−1
1 + xy x x2 y
= xy × = xy × = .
x 1 + xy 1 + xy
4.2. The Exponential Functions
Suppose that a ∈ R is a positive real number. We have shown in Section 4.1 that we can use (1)–(4)
to define ak for every rational number k ∈ Q. Here we shall briefly discuss how we may further extend
the definition of ak to a function ax defined for every real number x ∈ R. A thorough treatment of this
extension will require the study of the theory of continuous functions as well as the well known result
that the rational numbers are “dense” among the real numbers, and is beyond the scope of this set of
notes. We shall instead confine our discussion here to a heuristic treatment.
We all know simple functions like y = x2 (a parabola) or y = 2x + 3 (a straight line). It is possible
to draw the graph of such a function in one single stroke, without lifting our pen from the paper before
completing the drawing. Such functions are called continuous functions.
Suppose now that a positive real number a ∈ R has been chosen and fixed. Note that we have
already defined ak for every rational number k ∈ Q. We now draw the graph of a continuous function
on the xy-plane which will pass through every point (k, ak ) where k ∈ Q. It turns out that such a
function is unique. In other words, there is one and only one continuous function whose graph on the
xy-plane will pass through every point (k, ak ) where k ∈ Q. We call this function the exponential
function corresponding to the positive real number a, and write f (x) = ax .
5. Chapter 4 : Indices and Logarithms 4–5
LAWS FOR EXPONENTIAL FUNCTIONS. Suppose that a ∈ R is positive. Then a0 = 1. For
every x1 , x2 ∈ R, we have
(a) ax1 ax2 = ax1 +x2 ;
ax1
(b) x2 = ax1 −x2 ; and
a
(c) (ax1 )x2 = ax1 x2 .
(d) Furthermore, if a = 1, then ax1 = ax2 if and only if x1 = x2 .
Example 4.2.1. The graph of the exponential function y = 2x is shown below:
y
y = 2x
4
3
2
1
1/2
1/4
x
-2 -1 1 2
Example 4.2.2. The graph of the exponential function y = (1/3)x is shown below:
y
9
8
7
6
5
4
3
2 x
1
y=
1 3
x
-2 -1 1 2
Example 4.2.3. We have 8x × 24x = (23 )x × 24x = 23x 24x = 27x = (27 )x = 128x .
Example 4.2.4. We have 9x/2 × 27x/3 = (91/2 )x × (271/3 )x = 3x 3x = 32x = (32 )x = 9x .
6. 4–6 W W L Chen and X T Duong : Elementary Mathematics
Example 4.2.5. We have
32x+2 ÷ 82x−1 = 32x+2 × 81−2x = (25 )x+2 × (23 )1−2x = 25x+10 23−6x = 213−x .
Example 4.2.6. We have
212x
642x ÷ 162x = 212x ÷ 28x = = 24x = (24 )x = 16x .
28x
Example 4.2.7. We have
5x+1 + 5x−1 5x+1 + 5x−1 5x+1 + 5x−1 1
= = = .
5 x+2 + 5x 5×5 x+1 + 5 × 5x−1 5(5x+1 + 5x−1 ) 5
Example 4.2.8. We have
4x − 2x−1 22x − 2−1 2x 2x 2x − 1 2x 2x (2x − 1 )
= = 2
= 2
= 2x .
2x − 12 2x − 1
2 2x − 1
2 2x − 12
Example 4.2.9. Suppose that
1
25x = √ .
125
We can write 25x = (52 )x = 52x and
1 √
√ = ( 125)−1 = ((125)1/2 )−1 = ((53 )1/2 )−1 = 5−3/2 .
125
It follows that we must have 2x = −3/2, so that x = −3/4.
Example 4.2.10. Suppose that
2x−1
1
= 3(27−x ).
9
We can write 3(27−x ) = 3((33 )−x ) = 3 × 3−3x = 31−3x and
2x−1 2x−1
1 1
= = (3−2 )2x−1 = 32−4x .
9 32
It follows that we must have 1 − 3x = 2 − 4x, so that x = 1.
√ √
Example 4.2.11. Suppose that 9x = 3. We can write 9x = (32 )x = 32x and 3 = 31/2 . It follows
that we must have 2x = 1/2, so that x = 1/4.
Example 4.2.12. Suppose that 53x−4 = 1. We can write 1 = 50 . It follows that we must have
3x − 4 = 0, so that x = 4/3.
√
Example 4.2.13. Suppose that (0.125)x = 0.5. We can write
x x
1 1
(0.125)x = = = (2−3 )x = 2−3x
8 23
and
√ 1
1/2
0.5 = (0.5)1/2 = = (2−1 )1/2 = 2−1/2 .
2
It follows that we must have −3x = −1/2, so that x = 1/6.
7. Chapter 4 : Indices and Logarithms 4–7
Example 4.2.14. Suppose that 81−x × 2x−3 = 4. We can write 4 = 22 and
81−x × 2x−3 = (23 )1−x × 2x−3 = 23−3x × 2x−3 = 2−2x .
It follows that we must have −2x = 2, so that x = −1.
4.3. The Logarithmic Functions
The logarithmic functions are the inverses of the exponential functions. Suppose that a ∈ R is a positive
real number and a = 1. If y = ax , then x is called the logarithm of y to the base a, denoted by x = loga y.
In other words, we have
y = ax if and only if x = loga y.
For the special case when a = 10, we have the common logarithm log10 y of y. Another special case is
when a = e = 2.7182818 . . . , an irrational number. We have the natural logarithm loge y of y, sometimes
also denoted by log y or ln y. Whenever we mention a logarithmic function without specifying its base,
we shall assume that it is base e.
Remark. The choice of the number e is made to ensure that the derivative of the function ex is also
equal to ex for every x ∈ R. This is not the case for any other non-zero function, apart from constant
multiples of the function ex .
Example 4.3.1. The graph of the logarithmic function x = log y is shown below:
x
x = log y
1
y
1 e
LAWS FOR LOGARITHMIC FUNCTIONS. Suppose that a ∈ R is positive and a = 1. Then
loga 1 = 0 and loga a = 1. For every positive y1 , y2 ∈ R, we have
(a) loga (y1 y2 ) = loga y1 + loga y2 ;
y1
(b) loga = loga y1 − loga y2 ; and
y2
(c) loga (y1 ) = k loga y1 for every k ∈ R.
k
(d) Furthermore, loga y1 = loga y2 if and only if y1 = y2 .
INVERSE LAWS. Suppose that a ∈ R is positive and a = 1.
(a) For every real number x ∈ R, we have loga (ax ) = x.
(b) For every positive real number y ∈ R, we have aloga y = y.
8. 4–8 W W L Chen and X T Duong : Elementary Mathematics
Example 4.3.2. We have log2 8 = x if and only if 2x = 8, so that x = 3. Alternatively, we can use
one of the Inverse laws to obtain log2 8 = log2 (23 ) = 3.
Example 4.3.3. We have log5 125 = x if and only if 5x = 125, so that x = 3.
Example 4.3.4. We have log3 81 = x if and only if 3x = 81, so that x = 4.
Example 4.3.5. We have log10 1000 = x if and only if 10x = 1000, so that x = 3.
Example 4.3.6. We have
1 1
log4 =x if and only if 4x = .
8 8
This means that 22x = 2−3 , so that x = −3/2.
Example 4.3.7. We have
√
2 21/2 7
log2 = log2 = log2 (2−7/2 ) = − .
16 24 2
Example 4.3.8. We have
1 3
log3 √ = log3 (3−3/2 ) = − .
3 3 2
Example 4.3.9. Use your calculator to confirm that the following are correct to 3 decimal places:
1
log10 20 ≈ 1.301, log10 7 ≈ 0.845, log10 ≈ −0.698,
5
log 5 ≈ 1.609, log 21 ≈ 3.044, log 0.2 ≈ −1.609.
Example 4.3.10. We have
1 1 1
loga =− if and only if a−1/3 = .
3 3 3
It follows that
−3
1
a = (a−1/3 )−3 = = 33 = 27.
3
Example 4.3.11. We have
1 2 1
loga =− if and only if a−2/3 = .
4 3 4
It follows that
−3/2
1
a = (a−2/3 )−3/2 = = 43/2 = 8.
4
Example 4.3.12. We have
1
loga 4 = if and only if a1/2 = 4.
2
It follows that a = 16.
9. Chapter 4 : Indices and Logarithms 4–9
Example 4.3.13. Suppose that log5 y = −2. Then y = 5−2 = 1/25.
Example 4.3.14. Suppose that loga y = loga 3 + loga 5. Then since loga 3 + loga 5 = loga 15, we must
have y = 15.
Example 4.3.15. Suppose that loga y + 2 loga 4 = loga 20. Then
20
loga y = loga 20 − 2 loga 4 = loga 20 − loga (42 ) = loga 20 − loga 16 = loga ,
16
so that y = 20/16 = 5/4.
Example 4.3.16. Suppose that
1
log 6 − log y = log 12.
2
Then √
1 √ 6
log y = log 6 − log 12 = log( 6) − log 12 = log ,
2 12
√
so that y = 6/12.
For the next four examples, u and v are positive real numbers.
Example 4.3.17. We have loga (u10 ) ÷ loga u = 10 loga u ÷ loga u = 10.
Example 4.3.18. We have 5 loga u − loga (u5 ) = 5 loga u − 5 loga u = 0.
Example 4.3.19. We have
2 loga u + 2 loga v − loga ((uv)2 ) = loga (u2 ) + loga (v 2 ) − loga (u2 v 2 ) = loga (u2 v 2 ) − loga (u2 v 2 ) = 0.
Example 4.3.20. We have
u3 v √ u3 v
loga + loga ÷ loga ( u) = loga × ÷ loga (u1/2 )
v u v u
1
= loga (u2 ) ÷ loga (u1/2 ) = 2 loga u ÷ loga u = 4.
2
Example 4.3.21. Suppose that 2 log y = log(4 − 3y). Then since 2 log y = log(y 2 ), we must have
y 2 = 4 − 3y, so that y 2 + 3y − 4 = 0. This quadratic equation has roots
√
−3 ± 9 + 16
y= = 1 or − 4.
2
However, we have to discard the solution y = −4, since log(−4) is not defined. The only solution is
therefore y = 1.
√ √
Example 4.3.22. Suppose that log( y) = log y. Then
1
log y = log y, and so log y = 2 log y.
2
Squaring both sides and letting x = log y, we obtain the quadratic equation x2 = 4x, with solutions
x = 0 and x = 4. The equation log y = 0 corresponds to y = 1. The equation log y = 4 corresponds to
y = e4 .
10. 4–10 W W L Chen and X T Duong : Elementary Mathematics
Problems for Chapter 4
1. Simplify each of the following expressions:
a) 2162/3 b) 323/5 c) 64−1/6 d) 10000−3/4
2. Simplify each of the following expressions, where x denotes a positive real number:
1/2 −1/3
25 27
a) (5x2 )3 ÷ (25x−4 )1/2 b) (32x2 )2/5 × c) (16x12 )3/4 ×
x4 x6
−1/2
1 −1 (2x3 )1/2 (8x)1/2
d) (128x14 )−1/7 ÷ x e) 2
÷
9 (4x) (3x2 )3
3. Simplify each of the following expressions, where x and y denote suitable positive real numbers:
x−1 − y −1
a) (x2 y 3 )1/3 (x3 y 2 )−1/2 b) −3
x − y −3
−2 −2 −2 −2
x +y x −y
c) − d) xy ÷ (x−1 + y −1 )−1
x+y x−y
−1 3
1 6x−2
e) (x−2 − y −2 )(x − y)−1 (x−1 + y −1 )−1 f) (36x1/2 y 2 )3/2 ÷
xy y −2/3
4. Determine whether each of the following statements is correct:
√ √ √ √ √ √
a) 8 + 2 15 = 3 + 5 b) 16 − 4 15 = 6 − 10
5. Simplify each of the following expressions, where x denotes a positive real number:
2x+1 + 4x 9x − 4x
a) 32x × 23x b) 163x/4 ÷ 42x c) x−1 d) x
2 +1 3 + 2x
6. Solve each of the following equations:
x
1 1
a) 272−x = 9x−2 b) 4x = √ c) = 3 × 81−x
32 9
x
1
d) 2x × 16x = 4 × 8x e) 4x = 3x f) = 72x
7
7. Find the precise value of each of the following expressions:
1 √ 1
a) log3 b) log100 1000 c) log5 125 d) log4
27 32
8. Solve each of the following equations:
a) log2 y + 3 log2 (2y) = 3 b) log3 y = −3 c) log 7 − 2 √ y = 2 log 49
log
d) 2 log y = log(y + 2) log(7y − 12)
e) 2 log y = √ f) log y = 3 log y
g) 2 log y = log(y + 6) h) 2 log y = log y
9. Simplify each of the following expressions, where u, v and w denote suitable positive real numbers:
u4 v2 u
a) log 2 − log b) log(u3 v −6 ) − 3 log 2
v u v
uv uw vw
c) 3 log u − log(uv)3 + 3 log(vw) d) log + log + log − log(uvw)
w v u
u4
e) log 2 − log v + 2 log u ÷ (2 log u − log v) f) log(u2 − v 2 ) − log(u − v) − log(u + v)
v
u 3
g) log u3 + log v 2 − log − 5 log v h) 2 log(eu ) + log(ev ) − log(eu+v )
v
− ∗ − ∗ − ∗ − ∗ − ∗ −