 5.3.67: In Exercises 6784, condense the expression to the logarithmof a sin...
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 5.3.85: In Exercises 85 and 86, compare the logarithmic quantities. Iftwo a...
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 5.3.87: In Exercises 8790, use the followinginformation. The relationship b...
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 5.3.91: In Exercises 9194, find a logarithmicequation that relates and Expl...
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 5.3.95: GALLOPING SPEEDS OF ANIMALS Fourleggedanimals run with two differe...
 5.3.96: NAIL LENGTH The approximate lengths and diameters(in inches) of com...
 5.3.97: COMPARING MODELS A cup of water at an initialtemperature of is plac...
 5.3.98: PROOF Prove that logbuv logb u logb v.
 5.3.99: PROOF Prove that logb un n logb u.
 5.3.100: CAPSTONE A classmate claims that the followingare true.(a) lnu v ln...
 5.3.101: TRUE OR FALSE? In Exercises 101106, determinewhether the statement ...
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 5.3.107: In Exercises 107112, use the changeofbase formula torewrite the l...
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 5.3.113: THINK ABOUT IT Consider the functions below. f x lnx2, gx ln xln 2,...
 5.3.114: GRAPHICAL ANALYSIS Use a graphing utility tograph the functions giv...
 5.3.115: THINK ABOUT IT For how many integers between1 and 20 can the natura...
Solutions for Chapter 5.3: Properties of Logarithms
Full solutions for College Algebra  8th Edition
ISBN: 9781439048696
Solutions for Chapter 5.3: Properties of Logarithms
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Chapter 5.3: Properties of Logarithms includes 49 full stepbystep solutions. College Algebra was written by and is associated to the ISBN: 9781439048696. This textbook survival guide was created for the textbook: College Algebra , edition: 8. Since 49 problems in chapter 5.3: Properties of Logarithms have been answered, more than 29733 students have viewed full stepbystep solutions from this chapter.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

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.

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

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

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.

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

Outer product uv T
= column times row = rank one matrix.

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.