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# Solutions for Chapter 12.4: Exponential and Logarithmic Equations

## Full solutions for Introductory & Intermediate Algebra for College Students | 4th Edition

ISBN: 9780321758941

Solutions for Chapter 12.4: Exponential and Logarithmic Equations

Solutions for Chapter 12.4
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##### ISBN: 9780321758941

This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Introductory & Intermediate Algebra for College Students, edition: 4. Chapter 12.4: Exponential and Logarithmic Equations includes 162 full step-by-step solutions. Since 162 problems in chapter 12.4: Exponential and Logarithmic Equations have been answered, more than 68856 students have viewed full step-by-step solutions from this chapter. Introductory & Intermediate Algebra for College Students was written by and is associated to the ISBN: 9780321758941.

Key Math Terms and definitions covered in this textbook
• Augmented matrix [A b].

Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

• Back substitution.

Upper triangular systems are solved in reverse order Xn to Xl.

• Complete solution x = x p + Xn to Ax = b.

(Particular x p) + (x n in nullspace).

• Elimination.

A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

• Fast Fourier Transform (FFT).

A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn-1c can be computed with ne/2 multiplications. Revolutionary.

• Full row rank r = m.

Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

• Fundamental Theorem.

The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n - r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

• Hankel matrix H.

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

• Hermitian matrix A H = AT = A.

Complex analog a j i = aU of a symmetric matrix.

• Indefinite matrix.

A symmetric matrix with eigenvalues of both signs (+ and - ).

• Krylov subspace Kj(A, b).

The subspace spanned by b, Ab, ... , Aj-Ib. Numerical methods approximate A -I b by x j with residual b - Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

• Left nullspace N (AT).

Nullspace of AT = "left nullspace" of A because y T A = OT.

• Multiplicities AM and G M.

The algebraic multiplicity A M of A is the number of times A appears as a root of det(A - AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

• Partial pivoting.

In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

• Projection p = a(aTblaTa) onto the line through a.

P = aaT laTa has rank l.

• Skew-symmetric matrix K.

The transpose is -K, since Kij = -Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

• Solvable system Ax = b.

The right side b is in the column space of A.

• Spectral Theorem A = QAQT.

Real symmetric A has real A'S and orthonormal q's.

• Spectrum of A = the set of eigenvalues {A I, ... , An}.

Spectral radius = max of IAi I.

• Volume of box.

The rows (or the columns) of A generate a box with volume I det(A) I.

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