 4.1: In Exercises 15, graph f and g in the same rectangularcoordinate sy...
 4.2: In Exercises 15, graph f and g in the same rectangularcoordinate sy...
 4.3: In Exercises 15, graph f and g in the same rectangularcoordinate sy...
 4.4: In Exercises 15, graph f and g in the same rectangularcoordinate sy...
 4.5: In Exercises 15, graph f and g in the same rectangularcoordinate sy...
 4.6: In Exercises 69, find the domain of each function.f(x) = log3(x + 6)
 4.7: In Exercises 69, find the domain of each function.g(x) = log3 x + 6
 4.8: In Exercises 69, find the domain of each function.h(x) = log3(x + 6)2
 4.9: In Exercises 69, find the domain of each function.f(x) = 3x+6
 4.10: In Exercises 1020, evaluate each expression without using acalculat...
 4.11: In Exercises 1020, evaluate each expression without using acalculat...
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 4.19: In Exercises 1020, evaluate each expression without using acalculat...
 4.20: In Exercises 1020, evaluate each expression without using acalculat...
 4.21: In Exercises 2122, expand and evaluate numerical terms.log1xy1000
 4.22: In Exercises 2122, expand and evaluate numerical terms.ln(e19x20)
 4.23: In Exercises 2325, write each expression as a single logarithm.8 lo...
 4.24: In Exercises 2325, write each expression as a single logarithm.7 lo...
 4.25: In Exercises 2325, write each expression as a single logarithm.12ln...
 4.26: Use the formulasA = Pa1 + rn bntand A = Pertto solve this exercise....
 4.27: In Exercises 1929, evaluate each expression without using acalculat...
 4.28: In Exercises 1929, evaluate each expression without using acalculat...
 4.29: In Exercises 1929, evaluate each expression without using acalculat...
 4.30: Graph f(x) = 2x and g(x) = log2 x in the same rectangularcoordinate...
 4.31: Graph f(x) = 1 13 2 x and g(x) = log13x in the same rectangularcoor...
 4.32: In Exercises 3235, the graph of a logarithmic function is given.Sel...
 4.33: In Exercises 3235, the graph of a logarithmic function is given.Sel...
 4.34: In Exercises 3235, the graph of a logarithmic function is given.Sel...
 4.35: In Exercises 3235, the graph of a logarithmic function is given.Sel...
 4.36: In Exercises 3638, begin by graphing f(x) = log2 x. Then usetransfo...
 4.37: In Exercises 3638, begin by graphing f(x) = log2 x. Then usetransfo...
 4.38: In Exercises 3638, begin by graphing f(x) = log2 x. Then usetransfo...
 4.39: In Exercises 3940, graph f and g in the same rectangularcoordinate ...
 4.40: In Exercises 3940, graph f and g in the same rectangularcoordinate ...
 4.41: In Exercises 4143, find the domain of each logarithmic function.f(x...
 4.42: In Exercises 4143, find the domain of each logarithmic function.f(x...
 4.43: In Exercises 4143, find the domain of each logarithmic function.f(x...
 4.44: In Exercises 4446, use inverse properties of logarithms to simplify...
 4.45: In Exercises 4446, use inverse properties of logarithms to simplify...
 4.46: In Exercises 4446, use inverse properties of logarithms to simplify...
 4.47: On the Richter scale, the magnitude, R, of an earthquake ofintensit...
 4.48: Students in a psychology class took a final examination. Aspart of ...
 4.49: The formulat = 1clna AA  Nbdescribes the time, t, in weeks, that i...
 4.50: In Exercises 5053, use properties of logarithms to expand eachlogar...
 4.51: In Exercises 5053, use properties of logarithms to expand eachlogar...
 4.52: In Exercises 5053, use properties of logarithms to expand eachlogar...
 4.53: In Exercises 5053, use properties of logarithms to expand eachlogar...
 4.54: In Exercises 5457, use properties of logarithms to condense eachlog...
 4.55: In Exercises 5457, use properties of logarithms to condense eachlog...
 4.56: In Exercises 5457, use properties of logarithms to condense eachlog...
 4.57: In Exercises 5457, use properties of logarithms to condense eachlog...
 4.58: In Exercises 5859, use common logarithms or natural logarithmsand a...
 4.59: In Exercises 5859, use common logarithms or natural logarithmsand a...
 4.60: In Exercises 6063, determine whether each equation is true orfalse....
 4.61: In Exercises 6063, determine whether each equation is true orfalse....
 4.62: In Exercises 6063, determine whether each equation is true orfalse....
 4.63: In Exercises 6063, determine whether each equation is true orfalse....
 4.64: In Exercises 6473, solve each exponential equation. Wherenecessary,...
 4.65: In Exercises 6473, solve each exponential equation. Wherenecessary,...
 4.66: In Exercises 6473, solve each exponential equation. Wherenecessary,...
 4.67: In Exercises 6473, solve each exponential equation. Wherenecessary,...
 4.68: In Exercises 6473, solve each exponential equation. Wherenecessary,...
 4.69: In Exercises 6473, solve each exponential equation. Wherenecessary,...
 4.70: In Exercises 6473, solve each exponential equation. Wherenecessary,...
 4.71: In Exercises 6473, solve each exponential equation. Wherenecessary,...
 4.72: In Exercises 6473, solve each exponential equation. Wherenecessary,...
 4.73: In Exercises 6473, solve each exponential equation. Wherenecessary,...
 4.74: In Exercises 7479, solve each logarithmic equation.log4(3x  5) = 3
 4.75: In Exercises 7479, solve each logarithmic equation.3 + 4 ln(2x) = 15
 4.76: In Exercises 7479, solve each logarithmic equation.log2(x + 3) + lo...
 4.77: In Exercises 7479, solve each logarithmic equation.log3(x  1)  lo...
 4.78: In Exercises 7479, solve each logarithmic equation.ln(x + 4)  ln(x...
 4.79: In Exercises 7479, solve each logarithmic equation.log4(2x + 1) = l...
 4.80: The function P(x) = 14.7e0.21x models the averageatmospheric press...
 4.81: Newest Dinosaur: The PCosaurus? For the period from2009 through 20...
 4.82: 60 + Years after Brown v. Board In 1954, the SupremeCourt ended leg...
 4.83: Use the formula for compound interest with n compoundingsper year t...
 4.84: Use the formula for continuous compounding to solveExercises8485How...
 4.85: Use the formula for continuous compounding to solveExercises8485Wha...
 4.86: According to the U.S. Bureau of the Census, in 2000 therewere 35.3 ...
 4.87: Use the exponential decay model, A = A0ekt, to solve thisexercise. ...
 4.88: The functionf(t) = 500,0001 + 2499e0.92tmodels the number of peopl...
 4.89: Exercises 8991 present data inthe form of tables. For each dataset ...
 4.90: Exercises 8991 present data inthe form of tables. For each dataset ...
 4.91: Exercises 8991 present data inthe form of tables. For each dataset ...
 4.92: In Exercises 9293, rewrite the equation in terms of base e. Express...
 4.93: In Exercises 9293, rewrite the equation in terms of base e. Express...
 4.94: The figure on the right shows world population projectionsthrough t...
Solutions for Chapter 4: Exponential and Logarithmic Functions
Full solutions for College Algebra  7th Edition
ISBN: 9780134469164
Solutions for Chapter 4: Exponential and Logarithmic Functions
Get Full SolutionsSince 94 problems in chapter 4: Exponential and Logarithmic Functions have been answered, more than 32684 students have viewed full stepbystep solutions from this chapter. College Algebra was written by and is associated to the ISBN: 9780134469164. Chapter 4: Exponential and Logarithmic Functions includes 94 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: College Algebra , edition: 7.

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

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

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

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.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

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).

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

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

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

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·

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