 11.4.1: Graph each function defined in 18.
 11.4.2: Graph each function defined in 18.
 11.4.3: Graph each function defined in 18.
 11.4.4: Graph each function defined in 18.
 11.4.5: Graph each function defined in 18.
 11.4.6: Graph each function defined in 18.
 11.4.7: Graph each function defined in 18.
 11.4.8: Graph each function defined in 18.
 11.4.9: The scale of the graph shown in Figure 11.4.1 is onefourth inch to...
 11.4.10: a. Use the definition of logarithm to show that logbbx = x for all ...
 11.4.11: Let b > 1. a. Use the fact that u = logb v v = bu to show that a po...
 11.4.12: Give a graphical interpretation for property (11.4.2) in Example 11...
 11.4.13: Suppose a positive real number x satisfies the inequality 10m x < 1...
 11.4.14: a. Prove that if x is a positive real number and k is a nonnegative...
 11.4.15: If n is an odd integer and n > 1, is log2(n 1) = log2(n)? Justify y...
 11.4.16: If n is an odd integer and n > 1, is log2(n + 1) = log2(n)? Justify...
 11.4.17: If n is an odd integer and n > 1, is log2(n + 1) = log2(n)? Justify...
 11.4.18: In 18 and 19, indicate how many binary digits are needed to represe...
 11.4.19: In 18 and 19, indicate how many binary digits are needed to represe...
 11.4.20: It was shown in the text that the number of binary digits needed to...
 11.4.21: In each of 21 and 22, a sequence is specified by a recurrence relat...
 11.4.22: In each of 21 and 22, a sequence is specified by a recurrence relat...
 11.4.23: Define a sequence c1, c2, c3,..., recursively as follows: c1 = 0, c...
 11.4.24: Define a sequence c1, c2, c3,..., recursively as follows: c1 = 0, c...
 11.4.25: Exercises 2528 refer to properties 11.4.9 and 11.4.10. To solve the...
 11.4.26: Exercises 2528 refer to properties 11.4.9 and 11.4.10. To solve the...
 11.4.27: Exercises 2528 refer to properties 11.4.9 and 11.4.10. To solve the...
 11.4.28: Exercises 2528 refer to properties 11.4.9 and 11.4.10. To solve the...
 11.4.29: Use notation to express the following statement: x 2 7x 2 + 3x ...
 11.4.30: Derive each statement in 3033.
 11.4.31: Derive each statement in 3033.
 11.4.32: Derive each statement in 3033.
 11.4.33: Derive each statement in 3033.
 11.4.34: Show that 4n is not O(2n ).
 11.4.35: Prove each of the statements in 3540, assuming n is an integer vari...
 11.4.36: Prove each of the statements in 3540, assuming n is an integer vari...
 11.4.37: Prove each of the statements in 3540, assuming n is an integer vari...
 11.4.38: Prove each of the statements in 3540, assuming n is an integer vari...
 11.4.39: Prove each of the statements in 3540, assuming n is an integer vari...
 11.4.40: Prove each of the statements in 3540, assuming n is an integer vari...
 11.4.41: Quantities of the form kn + kn log2 n for positive integers k1 k2, ...
 11.4.42: Calculate the values of the harmonic sums 1 + 1 2 + 1 3 ++ 1 n for ...
 11.4.43: Use part (d) of Example 11.4.7 to show that n + n 2 + n 3 ++ n n is...
 11.4.44: Use the fact that log2 x = 1 loge 2 loge x and loge x = ln x, for a...
 11.4.45: a. Show that log2 n is (log2 n). b. Show that log2 n + 1 is (log2 n...
 11.4.46: Prove by mathematical induction that n 10n for all integers n 1.
 11.4.47: Prove by mathematical induction that log2 n n for all integers n 1.
 11.4.48: Show that if n is a variable that takes positive integer values, th...
 11.4.49: Let n be a variable that takes positive integer values. a. Show tha...
 11.4.50: a. For all positive real numbers u, log2 u < u. Use this fact to sh...
 11.4.51: a. For all real numbers x, x < 2x . Use this fact to show that for ...
 11.4.52: For all positive real numbers u, log2 u < u. Use this fact and the ...
 11.4.53: Use the result of exercise 52 above to prove the following: For all...
 11.4.54: Exercises 54 and 55 use LHpitals rule from calculus.
 11.4.55: Exercises 54 and 55 use LHpitals rule from calculus.
 11.4.56: Complete the proof in Example 11.4.4.
Solutions for Chapter 11.4: Exponential and Logarithmic Functions: Graphs and Orders
Full solutions for Discrete Mathematics with Applications  4th Edition
ISBN: 9780495391326
Solutions for Chapter 11.4: Exponential and Logarithmic Functions: Graphs and Orders
Get Full SolutionsThis textbook survival guide was created for the textbook: Discrete Mathematics with Applications , edition: 4th. This expansive textbook survival guide covers the following chapters and their solutions. Since 56 problems in chapter 11.4: Exponential and Logarithmic Functions: Graphs and Orders have been answered, more than 25636 students have viewed full stepbystep solutions from this chapter. Chapter 11.4: Exponential and Logarithmic Functions: Graphs and Orders includes 56 full stepbystep solutions. Discrete Mathematics with Applications was written by and is associated to the ISBN: 9780495391326.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

Column space C (A) =
space of all combinations of the columns of A.

Dimension of vector space
dim(V) = number of vectors in any basis for V.

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.