Lecture 12 sections 4.5 logarithmic equations
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This document covers solving logarithmic and exponential equations. It begins with examples of solving exponential equations by isolating the exponential term, taking logarithms of both sides, and then solving for the variable. Subsequent examples show solving equations with different bases by first rewriting them in exponential form using logarithm properties. The last example demonstrates using the product and quotient rules for logarithms to solve equations.
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This document provides an introduction to solving equations involving indices and logarithms. It begins with examples of rewriting logarithmic expressions in exponential form and vice versa. The key rule discussed is that rewriting problems can help when getting stuck. Examples are then provided of solving logarithmic equations by setting the arguments equal when the bases are the same. It concludes by explaining that logarithmic expressions with negative bases are undefined since there is no exponential that equals a negative number.
Unit 2.7
This document discusses solving rational equations and eliminating extraneous solutions. It begins by explaining that rational functions are often used as models that require solving equations involving fractions. When multiplying or dividing terms, extra solutions can arise that do not satisfy the original equation. Examples are provided to demonstrate solving rational equations by clearing fractions and identifying extraneous solutions. One example solves an equation to find the minimum perimeter of a rectangle with a given area.
Similar to Lecture 12 sections 4.5 logarithmic equations (20)
Section 14.6 solving equations with rational expressions
Section 14.6 solving equations with rational expressions
Chapter12 additionaldifferentiationtopics-151003154510-lva1-app6891
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Lecture 13 gram-schmidt inner product spaces - 6.4 6.7
The document discusses the Gram-Schmidt process for finding an orthogonal basis. It presents the Gram-Schmidt algorithm in 3 steps: 1) the first vector in the orthogonal basis is the first vector in the original basis, 2) subsequent vectors are found by removing projections onto previous vectors, 3) repeat step 2 for all other vectors. It also provides an example applying Gram-Schmidt to find an orthogonal basis for the space spanned by three vectors.
Lecture 12 orhogonality - 6.1 6.2 6.3
1. The orthogonal decomposition theorem states that any vector y in Rn can be written uniquely as the sum of a vector ŷ in a subspace W and a vector z orthogonal to W.
2. The vector ŷ is called the orthogonal projection of y onto W. It is the closest vector to y that lies in W.
3. The best approximation theorem states that the orthogonal projection ŷ provides the best or closest approximation of y using only vectors that lie in the subspace W. The distance from y to ŷ is less than the distance from y to any other vector in
Lecture 11 diagonalization & complex eigenvalues - 5-3 & 5-5
This document discusses diagonalization of matrices. It defines similarity of matrices and notes that similar matrices have the same characteristic polynomial and eigenvalues. It then discusses diagonalizing matrices by finding the eigenvalues and corresponding eigenvectors, constructing a change of basis matrix P from the eigenvectors, and constructing a diagonal matrix D from the eigenvalues. It provides examples of diagonalizing matrices with real and complex eigenvalues.
Lecture 9 eigenvalues - 5-1 & 5-2
This document discusses eigenvalues and eigenvectors. It defines an eigenvector as a nonzero vector that does not change direction when a matrix is multiplied by it. An eigenvalue is a scalar value such that this equation is satisfied. The document provides examples of finding eigenvalues and eigenvectors. It also discusses eigenspaces, which are subspaces containing eigenvectors. Additionally, it covers the characteristic equation for finding eigenvalues of a matrix and discusses similarity of matrices.
Lecture 9 dim & rank - 4-5 & 4-6
The document discusses the dimension of vector spaces. It defines the dimension of a vector space V as the number of vectors in a basis for V. It states that any set containing more vectors than the dimension must be linearly dependent, and that every basis of V must consist of exactly the dimension number of vectors. Examples of finding the dimension of various vector spaces and subspaces are provided.
Lecture 8 nul col bases dim & rank - section 4-2, 4-3, 4-5 & 4-6
The document discusses null spaces, column spaces, and bases of matrices. It begins by defining the null space of a matrix A as the set of all solutions to the homogeneous equation Ax = 0. It then proves that the null space of any matrix is a subspace. Similarly, it defines the column space of A as the set of all linear combinations of A's columns, and proves the column space is always a subspace. The document contrasts the properties of null spaces and column spaces. It also discusses finding bases for null spaces and column spaces. Finally, it covers linear independence, spanning sets, and using pivots to determine bases.
Lecture 7 determinants cramers spaces - section 3-2 3-3 and 4-1
The document discusses properties of determinants, row operations on matrices, and how they affect determinants. It then covers Cramer's rule, vector spaces, subspaces, and the null space and column space of matrices. Specifically, it provides theorems showing that row replacements and scalings of rows do not change the determinant, while row interchanges negate the determinant. Cramer's rule is introduced for solving systems of linear equations. Key concepts for vector spaces and subspaces are defined, and the null space and column space of a matrix are shown to be subspaces.
Lecture 6 lu factorization & determinants - section 2-5 2-7 3-1 and 3-2
The document discusses LU factorization for solving systems of linear equations Ax = b. It explains that LU factorization factors a matrix A into lower and upper triangular matrices L and U, and that this can be used to solve systems of linear equations by first solving Ly = b for y, and then solving Ux = y for x. It also briefly discusses computer graphics topics like translations using homogeneous coordinates, and properties of determinants such as cofactor expansion and how row operations affect the determinant.
Lecture 5 inverse of matrices - section 2-2 and 2-3
The document discusses the inverse of matrices. It defines an inverse matrix A-1 as a matrix such that AA-1=I=A-1A, where I is the identity matrix. It provides methods for determining if a 2x2 matrix is invertible based on its determinant, and for finding the inverse of a matrix using elementary row operations. It also presents the Invertible Matrix Theorem, which provides equivalent conditions for a matrix to be invertible.
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This document defines common matrix notation and operations such as matrix addition, scalar multiplication, and matrix multiplication. It also covers properties of these operations, as well as the transpose and powers of a matrix. Key concepts include representing a matrix as a collection of columns representing vectors, defining entries and dimensions of a matrix, and rules for matrix algebra like the associative and distributive properties.
Lecture 4 chapter 1 review section 2-1
This document defines common matrix notation and operations. It discusses that a matrix A with m rows and n columns has entries aij. The columns of A are vectors that form the matrix. Matrix addition and scalar multiplication are defined element-wise. Matrix multiplication is defined as the product of the columns of the first matrix with the rows of the second. Properties of matrix addition, multiplication and transposition are provided.
Lecture 3 section 1-7, 1-8 and 1-9
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Lecture 02
1. The document discusses properties of vectors and systems of linear equations, including the parallelogram rule for vector addition, properties of linear combinations of vectors, homogeneous and nonhomogeneous systems, and writing solutions to consistent systems in parametric vector form.
2. Key concepts covered include the definition of the zero vector, properties of vector addition and scalar multiplication, the relationship between matrix-vector equations and systems of linear equations, and determining whether a vector can be written as a linear combination of other vectors.
3. Methods are presented for determining if a homogeneous system has nontrivial solutions, describing the solution set of a homogeneous system, and writing the solution set of a consistent nonhomogeneous system in parametric vector form
Lecture 01 - Section 1.1 & 1.2 Row Operations & Row Reduction
The document discusses elementary row operations and row echelon form (REF) and reduced row echelon form (RREF) for matrices. It describes the row reduction algorithm to convert a matrix into REF and RREF forms using elementary row operations. The algorithm proceeds by finding the leftmost nonzero pivot column at each step and using row operations to create zeros below the pivot. Converting a matrix to RREF form allows determining whether a linear system has no solution, a unique solution, or infinitely many solutions.
Lecture 01 - Row Operations & Row Reduction
The document discusses elementary row operations and row echelon form (REF) and reduced row echelon form (RREF) for solving systems of linear equations. It describes the row reduction algorithm which uses row operations to systematically reduce a matrix to REF or RREF form. The algorithm proceeds by finding the leftmost nonzero pivot column, then uses row operations to create zeros below the pivot entry to put the matrix in echelon form. Working backward and upward, it further reduces the matrix to RREF form. From the RREF form, the existence and number of solutions for the corresponding linear system can be determined.
Lecture 20 fundamental theorem of calc - section 5.3
The document discusses indefinite integrals and introduces the notation used to represent them. It explains that an indefinite integral of a function f(x) is represented by ∫f(x) dx and denotes the general antiderivative of f(x), not a specific function. The document also provides an example of finding the indefinite integral of 10x4 - 2sec2x and explains how to check the answer.
Lecture 18 antiderivatives - section 4.8
This document discusses antiderivatives and provides examples of finding antiderivatives of specific functions. It defines antiderivatives and notes their general form involves an arbitrary constant C. Example 1 finds the antiderivatives of sin x, 1/x, and x^n where n ≠ -1. It uses properties of derivatives and integral rules like the power rule to determine the antiderivatives are -cos x + C, ln|x| + C, and (x^(n+1))/(n+1) + C respectively. The document continues explaining using initial conditions to determine the value of C.
Lecture 17 optimization - section 4.6
The document discusses optimization problems involving maximizing or minimizing an objective function subject to constraints. It provides examples of maximizing the area of a fenced field given a fixed fence length, and outlines the general process of solving optimization problems by identifying variables, deriving the objective function, taking derivatives to find critical points, and applying tests like the maximum/minimum interval test or second derivative test to determine if a critical point is maximum or minimum. It also provides exercises asking to maximize/minimize things like perimeter given an area constraint, volume of an open box constructed from a square sheet, and area of a rectangle with vertices on a parabola.
Lecture 16 graphing - section 4.3
This document discusses concepts related to derivatives and curves, including:
- Increasing and decreasing functions and the first derivative test for local maxima and minima.
- Concavity, defined as whether a curve is concave upward or downward. The second derivative test can determine concavity.
- Points of inflection, where a curve changes from being concave upward to downward or vice versa.
It then provides an example problem analyzing the curve y=x4-4x3, using derivatives to find critical points and concavity, identify points of inflection, and sketch the curve.
Lecture 15 max min - section 4.2
This document discusses maximum and minimum values of functions. It defines absolute (global) and relative (local) extremes. The Extreme Value Theorem states that a continuous function on a closed interval will attain both a maximum and minimum value. However, these extremes may not exist if the function is not continuous or if the domain is not a closed interval. To find extremes, we look at critical points where the derivative is zero or undefined and the endpoints.
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Lecture 12 sections 4.5 logarithmic equations
- 1. MATH 107 Section 4.5 Logarithmic and Exponential Equations
- 2. 2© 2010 Pearson Education, Inc. All rights reserved EXAMPLE 1 Solving an Exponential Equation a. 25x = 125 Solve each equation. b. 9x = 3x+1 Solution
- 3. 3© 2010 Pearson Education, Inc. All rights reserved OBJECTIVE Solve exponential equations when both sides are not expressed with the same base. Step 1 Isolate the exponential expression on one side of the equation. Step 2 Take the common or natural logarithm of both sides. Step 3 Use the power rule, loga M r = r loga M. EXAMPLE 2 Solving Exponential Equations Using the Logarithms EXAMPLE Solve for x: 5 ∙ 2x – 3 = 17.
- 4. 4© 2010 Pearson Education, Inc. All rights reserved OBJECTIVE Solve exponential equations when both sides are not expressed with the same base. Step 4 Solve for the variable. EXAMPLE 2 Solving Exponential Equations Using the Logarithms EXAMPLE Solve for x: 5 ∙ 2x – 3 = 17.
- 5. Practice Problem 5© 2010 Pearson Education, Inc. All rights reserved
- 6. 6© 2010 Pearson Education, Inc. All rights reserved EXAMPLE 3 Solving an Exponential Equation with Different Bases ( ) ( ) ( ) 2 3 1 ln5 ln3 ln5 ln3 2 ln5 3ln5 ln3 ln3 2 ln5 ln3 ln3 3ln5 2ln5 ln3 ln3 3ln5 ln3 3ln5 2.795 2ln5 3 n3 2 1 l x x x x x x x x x x − + = = − = + − = + − = + + = ≈ − − + Solve the equation 52x–3 = 3x+1 and approximate the answer to three decimal places. When different bases are involved, begin with Step 2. Solution
- 7. Practice Problem 7© 2010 Pearson Education, Inc. All rights reserved
- 8. 8© 2010 Pearson Education, Inc. All rights reserved SOLVING LOGARITHMS EQUATIONS 4 2log means 2 64 1x x= = = Equations that contain terms of the form log a x are called logarithmic equations. To solve a logarithmic equation we write it in the equivalent exponential form. log2 x = 4 log3 2x −1( ) = log3 x + 2( ) log2 x − 3( ) + log2 x − 4( ) = 1
- 9. 9© 2010 Pearson Education, Inc. All rights reserved EXAMPLE 7 Solving a Logarithmic Equation Solve: 4 + 3log2 x = 1. We must check our solution. Solution 2 2 2 1 4 3log 1 3log 1 4 3 log 1 2 1 2 x x x x x − + = = − = − = − = =
- 10. 10© 2010 Pearson Education, Inc. All rights reserved EXAMPLE 7 Solving a Logarithmic Equation Solution continued 1 2 .The solution set is 1 2 .Check x = ? 2 ? 2 ? 2 ? ? 1 4 3log 1 1 4 3log 1 2 4 3log 1 4 3 2 1 1 1 x − + = + = + = − = =
- 11. Practice Problem 11© 2010 Pearson Education, Inc. All rights reserved
- 12. 12© 2010 Pearson Education, Inc. All rights reserved EXAMPLE 9 Using the Product and Quotient Rules Solve: ( ) ( )2 2a. log 3 log 4 1x x− + − = ( ) ( )2 2b. log 4 log 3 1.x x+ − + = Solution ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 1 2 2 a. log 3 log 4 1 log 3 4 1 3 4 2 7 12 2 7 10 0 2 5 0 x x x x x x x x x x x x − + − = − − = − − = − + = − + = − − =
- 13. 13© 2010 Pearson Education, Inc. All rights reserved EXAMPLE 9 Using the Product and Quotient Rules Solution continued x − 2 = 0 or x − 5 = 0 x = 2 or x = 5 Check x = 2 Logarithms are not defined for negative numbers, so x = 2 is not a solution. ( ) ( ) ( ) ( ) 2 2 2 2 log 3 log 4 1 log 1 lo 2 2 g 2 1 − + − = − + − = ? ?
- 14. 14© 2010 Pearson Education, Inc. All rights reserved EXAMPLE 9 Using the Product and Quotient Rules Solution continued The solution set is {5}. Check x = 5 ( ) ( ) ( ) ( ) 2 2 2 2 log 3 log 4 1 log 2 log 1 1 1 0 1 1 1 5 5− + − = + = + = = ? ? ?
- 15. 15© 2010 Pearson Education, Inc. All rights reserved EXAMPLE 9 Using the Product and Quotient Rules Solution continued b.
- 16. 16© 2010 Pearson Education, Inc. All rights reserved EXAMPLE 9 Using the Product and Quotient Rules Solution continued Check x = −2 1 1= Check x = −5 log2 (−1) and log2 (−2) are undefined, so solution set is {− 2}.
- 17. Practice Problem 17© 2010 Pearson Education, Inc. All rights reserved