 12.4.12.1.359: Fill in each blank so that the resulting statement is true. If bM b...
 12.4.12.1.360: Fill in each blank so that the resulting statement is true. If 24x ...
 12.4.12.1.361: Fill in each blank so that the resulting statement is true. If x ln...
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 12.4.12.1.369: Solve each exponential equation in Exercises 118 by expressing each...
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 12.4.12.1.387: Solve each exponential equation in Exercises 1940 by taking the log...
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 12.4.12.1.409: Solve each logarithmic equation in Exercises 4190. Be sure to rejec...
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 12.4.12.1.459: In Exercises 9198, solve each equation. 52x 54x 125
 12.4.12.1.460: In Exercises 9198, solve each equation. 3x 2 3x 81
 12.4.12.1.461: In Exercises 9198, solve each equation. 3x2 45
 12.4.12.1.462: In Exercises 9198, solve each equation. 5x2 50
 12.4.12.1.463: In Exercises 9198, solve each equation. log2(x 6) log2(x 4) log2 x 2
 12.4.12.1.464: In Exercises 9198, solve each equation. log2(x 3) log2 x log2(x 2) 2
 12.4.12.1.465: In Exercises 9198, solve each equation. 5x2 12 252x
 12.4.12.1.466: In Exercises 9198, solve each equation. 3x2 12 92x
 12.4.12.1.467: The formula A 36.1e0.0126t models the population of California, A, ...
 12.4.12.1.468: The formula A 22.9e0.0183t models the population of Texas, A, in mi...
 12.4.12.1.469: The function f (x) 20(0.975)x models the percentage of surface sunl...
 12.4.12.1.470: The function f (x) 20(0.975)x models the percentage of surface sunl...
 12.4.12.1.471: In Exercises 103106, complete the table for a savings account subje...
 12.4.12.1.472: In Exercises 103106, complete the table for a savings account subje...
 12.4.12.1.473: In Exercises 103106, complete the table for a savings account subje...
 12.4.12.1.474: In Exercises 103106, complete the table for a savings account subje...
 12.4.12.1.475: In Exercises 107110, complete the table for a savings account subje...
 12.4.12.1.476: In Exercises 107110, complete the table for a savings account subje...
 12.4.12.1.477: In Exercises 107110, complete the table for a savings account subje...
 12.4.12.1.478: In Exercises 107110, complete the table for a savings account subje...
 12.4.12.1.479: The data can be modeled by the function f (x) 1.2 lnx 15.7, where f...
 12.4.12.1.480: The data can be modeled by the function f (x) 1.2 lnx 15.7, where f...
 12.4.12.1.481: The function P(x) 95 30 log2 x models the percentage, P(x), of stud...
 12.4.12.1.482: The function P(x) 95 30 log2 x models the percentage, P(x), of stud...
 12.4.12.1.483: The pH of a solution is given by pH log x, where x represents the c...
 12.4.12.1.484: The pH of a solution is given by pH log x, where x represents the c...
 12.4.12.1.485: What is an exponential equation?
 12.4.12.1.486: Explain how to solve an exponential equation when both sides can be...
 12.4.12.1.487: Explain how to solve an exponential equation when both sides cannot...
 12.4.12.1.488: What is a logarithmic equation?
 12.4.12.1.489: Explain the differences between solving log3(x 1) 4 and log3(x 1) l...
 12.4.12.1.490: In many states, a 17% risk of a car accident with a blood alcohol c...
 12.4.12.1.491: In Exercises 123130, use your graphing utility to graph each side o...
 12.4.12.1.492: In Exercises 123130, use your graphing utility to graph each side o...
 12.4.12.1.493: In Exercises 123130, use your graphing utility to graph each side o...
 12.4.12.1.494: In Exercises 123130, use your graphing utility to graph each side o...
 12.4.12.1.495: In Exercises 123130, use your graphing utility to graph each side o...
 12.4.12.1.496: In Exercises 123130, use your graphing utility to graph each side o...
 12.4.12.1.497: In Exercises 123130, use your graphing utility to graph each side o...
 12.4.12.1.498: In Exercises 123130, use your graphing utility to graph each side o...
 12.4.12.1.499: Hurricanes are one of natures most destructive forces. These lowpr...
 12.4.12.1.500: Hurricanes are one of natures most destructive forces. These lowpr...
 12.4.12.1.501: The function P(t) 145e0.092t models a runners pulse, P(t), in beats...
 12.4.12.1.502: The function W(t) 2600(1 0.51e0.075t)3 models the weight, W(t), in ...
 12.4.12.1.503: In Exercises 135138, determine whether each statement makes sense o...
 12.4.12.1.504: In Exercises 135138, determine whether each statement makes sense o...
 12.4.12.1.505: In Exercises 135138, determine whether each statement makes sense o...
 12.4.12.1.506: In Exercises 135138, determine whether each statement makes sense o...
 12.4.12.1.507: In Exercises 139142, determine whether each statement is true or fa...
 12.4.12.1.508: In Exercises 139142, determine whether each statement is true or fa...
 12.4.12.1.509: In Exercises 139142, determine whether each statement is true or fa...
 12.4.12.1.510: In Exercises 139142, determine whether each statement is true or fa...
 12.4.12.1.511: If $4000 is deposited into an account paying 3% interest compounded...
 12.4.12.1.512: Solve each equation in Exercises 144146. Check each proposed soluti...
 12.4.12.1.513: Solve each equation in Exercises 144146. Check each proposed soluti...
 12.4.12.1.514: Solve each equation in Exercises 144146. Check each proposed soluti...
 12.4.12.1.515: Solve: 2x 1 x 1 1. (Section 10.6, Example 4)
 12.4.12.1.516: Solve: 3 x 1 5 x 19 x2 x . (Section 7.6, Example 4)
 12.4.12.1.517: Simplify: (2x3y2)4. (Section 5.7, Example 6)
 12.4.12.1.518: Exercises 150152 will help you prepare for the material covered in ...
 12.4.12.1.519: Exercises 150152 will help you prepare for the material covered in ...
 12.4.12.1.520: Exercises 150152 will help you prepare for the material covered in ...
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
Get Full SolutionsThis 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 stepbystep solutions. Since 162 problems in chapter 12.4: Exponential and Logarithmic Equations have been answered, more than 68856 students have viewed full stepbystep solutions from this chapter. Introductory & Intermediate Algebra for College Students was written by and is associated to the ISBN: 9780321758941.

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 Fn1c 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, ... , AjIb. 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.

Skewsymmetric 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.