 4.3.1: In Exercises 1 and 2, fill in the blank(s). 1. You can evaluate log...
 4.3.2: In Exercises 1 and 2, fill in the blank(s). 2. Two properties of lo...
 4.3.3: Is log3 24 = ln 3 ln 24 or log3 24 = ln 24 ln 3 correct?
 4.3.4: Which property of logarithms can you use to condense the expression...
 4.3.5: In Exercises 512, rewrite the logarithm as a ratio of (a) common lo...
 4.3.6: In Exercises 512, rewrite the logarithm as a ratio of (a) common lo...
 4.3.7: In Exercises 512, rewrite the logarithm as a ratio of (a) common lo...
 4.3.8: In Exercises 512, rewrite the logarithm as a ratio of (a) common lo...
 4.3.9: In Exercises 512, rewrite the logarithm as a ratio of (a) common lo...
 4.3.10: In Exercises 512, rewrite the logarithm as a ratio of (a) common lo...
 4.3.11: In Exercises 512, rewrite the logarithm as a ratio of (a) common lo...
 4.3.12: In Exercises 512, rewrite the logarithm as a ratio of (a) common lo...
 4.3.13: In Exercises 1320, evaluate the logarithm using the changeofbase ...
 4.3.14: In Exercises 1320, evaluate the logarithm using the changeofbase ...
 4.3.15: In Exercises 1320, evaluate the logarithm using the changeofbase ...
 4.3.16: In Exercises 1320, evaluate the logarithm using the changeofbase ...
 4.3.17: In Exercises 1320, evaluate the logarithm using the changeofbase ...
 4.3.18: In Exercises 1320, evaluate the logarithm using the changeofbase ...
 4.3.19: In Exercises 1320, evaluate the logarithm using the changeofbase ...
 4.3.20: In Exercises 1320, evaluate the logarithm using the changeofbase ...
 4.3.21: In Exercises 2124, rewrite the expression in terms of ln 4 and ln 5...
 4.3.22: In Exercises 2124, rewrite the expression in terms of ln 4 and ln 5...
 4.3.23: In Exercises 2124, rewrite the expression in terms of ln 4 and ln 5...
 4.3.24: In Exercises 2124, rewrite the expression in terms of ln 4 and ln 5...
 4.3.25: In Exercises 2528, approximate the logarithm using the properties o...
 4.3.26: In Exercises 2528, approximate the logarithm using the properties o...
 4.3.27: In Exercises 2528, approximate the logarithm using the properties o...
 4.3.28: In Exercises 2528, approximate the logarithm using the properties o...
 4.3.29: In Exercises 2936, use the changeofbase formula loga x = (ln x)(l...
 4.3.30: In Exercises 2936, use the changeofbase formula loga x = (ln x)(l...
 4.3.31: In Exercises 2936, use the changeofbase formula loga x = (ln x)(l...
 4.3.32: In Exercises 2936, use the changeofbase formula loga x = (ln x)(l...
 4.3.33: In Exercises 2936, use the changeofbase formula loga x = (ln x)(l...
 4.3.34: In Exercises 2936, use the changeofbase formula loga x = (ln x)(l...
 4.3.35: In Exercises 2936, use the changeofbase formula loga x = (ln x)(l...
 4.3.36: In Exercises 2936, use the changeofbase formula loga x = (ln x)(l...
 4.3.37: In Exercises 3744, use the properties of logarithms to rewrite and ...
 4.3.38: In Exercises 3744, use the properties of logarithms to rewrite and ...
 4.3.39: In Exercises 3744, use the properties of logarithms to rewrite and ...
 4.3.40: In Exercises 3744, use the properties of logarithms to rewrite and ...
 4.3.41: In Exercises 3744, use the properties of logarithms to rewrite and ...
 4.3.42: In Exercises 3744, use the properties of logarithms to rewrite and ...
 4.3.43: In Exercises 3744, use the properties of logarithms to rewrite and ...
 4.3.44: In Exercises 3744, use the properties of logarithms to rewrite and ...
 4.3.45: In Exercises 45 and 46, use the properties of logarithms to verify ...
 4.3.46: In Exercises 45 and 46, use the properties of logarithms to verify ...
 4.3.47: In Exercises 4764, use the properties of logarithms to expand the e...
 4.3.48: In Exercises 4764, use the properties of logarithms to expand the e...
 4.3.49: In Exercises 4764, use the properties of logarithms to expand the e...
 4.3.50: In Exercises 4764, use the properties of logarithms to expand the e...
 4.3.51: In Exercises 4764, use the properties of logarithms to expand the e...
 4.3.52: In Exercises 4764, use the properties of logarithms to expand the e...
 4.3.53: In Exercises 4764, use the properties of logarithms to expand the e...
 4.3.54: In Exercises 4764, use the properties of logarithms to expand the e...
 4.3.55: In Exercises 4764, use the properties of logarithms to expand the e...
 4.3.56: In Exercises 4764, use the properties of logarithms to expand the e...
 4.3.57: In Exercises 4764, use the properties of logarithms to expand the e...
 4.3.58: In Exercises 4764, use the properties of logarithms to expand the e...
 4.3.59: In Exercises 4764, use the properties of logarithms to expand the e...
 4.3.60: In Exercises 4764, use the properties of logarithms to expand the e...
 4.3.61: In Exercises 4764, use the properties of logarithms to expand the e...
 4.3.62: In Exercises 4764, use the properties of logarithms to expand the e...
 4.3.63: In Exercises 4764, use the properties of logarithms to expand the e...
 4.3.64: In Exercises 4764, use the properties of logarithms to expand the e...
 4.3.65: In Exercises 65 68, (a) use a graphing utility to graph the two equ...
 4.3.66: In Exercises 65 68, (a) use a graphing utility to graph the two equ...
 4.3.67: In Exercises 65 68, (a) use a graphing utility to graph the two equ...
 4.3.68: In Exercises 65 68, (a) use a graphing utility to graph the two equ...
 4.3.69: In Exercises 69 84, use the properties of logarithms to condense th...
 4.3.70: In Exercises 69 84, use the properties of logarithms to condense th...
 4.3.71: In Exercises 69 84, use the properties of logarithms to condense th...
 4.3.72: In Exercises 69 84, use the properties of logarithms to condense th...
 4.3.73: In Exercises 69 84, use the properties of logarithms to condense th...
 4.3.74: In Exercises 69 84, use the properties of logarithms to condense th...
 4.3.75: In Exercises 69 84, use the properties of logarithms to condense th...
 4.3.76: In Exercises 69 84, use the properties of logarithms to condense th...
 4.3.77: In Exercises 69 84, use the properties of logarithms to condense th...
 4.3.78: In Exercises 69 84, use the properties of logarithms to condense th...
 4.3.79: In Exercises 69 84, use the properties of logarithms to condense th...
 4.3.80: In Exercises 69 84, use the properties of logarithms to condense th...
 4.3.81: In Exercises 69 84, use the properties of logarithms to condense th...
 4.3.82: In Exercises 69 84, use the properties of logarithms to condense th...
 4.3.83: In Exercises 69 84, use the properties of logarithms to condense th...
 4.3.84: In Exercises 69 84, use the properties of logarithms to condense th...
 4.3.85: In Exercises 85 88, (a) use a graphing utility to graph the two equ...
 4.3.86: In Exercises 85 88, (a) use a graphing utility to graph the two equ...
 4.3.87: In Exercises 85 88, (a) use a graphing utility to graph the two equ...
 4.3.88: In Exercises 85 88, (a) use a graphing utility to graph the two equ...
 4.3.89: In Exercises 89 102, find the exact value of the logarithm without ...
 4.3.90: In Exercises 89 102, find the exact value of the logarithm without ...
 4.3.91: In Exercises 89 102, find the exact value of the logarithm without ...
 4.3.92: In Exercises 89 102, find the exact value of the logarithm without ...
 4.3.93: In Exercises 89 102, find the exact value of the logarithm without ...
 4.3.94: In Exercises 89 102, find the exact value of the logarithm without ...
 4.3.95: In Exercises 89 102, find the exact value of the logarithm without ...
 4.3.96: In Exercises 89 102, find the exact value of the logarithm without ...
 4.3.97: In Exercises 89 102, find the exact value of the logarithm without ...
 4.3.98: In Exercises 89 102, find the exact value of the logarithm without ...
 4.3.99: In Exercises 89 102, find the exact value of the logarithm without ...
 4.3.100: In Exercises 89 102, find the exact value of the logarithm without ...
 4.3.101: In Exercises 89 102, find the exact value of the logarithm without ...
 4.3.102: In Exercises 89 102, find the exact value of the logarithm without ...
 4.3.103: In Exercises 103 106, (a) use a graphing utility to graph the two e...
 4.3.104: In Exercises 103 106, (a) use a graphing utility to graph the two e...
 4.3.105: In Exercises 103 106, (a) use a graphing utility to graph the two e...
 4.3.106: In Exercises 103 106, (a) use a graphing utility to graph the two e...
 4.3.107: The relationship between the number of decibels and the intensity o...
 4.3.108: The table shows the approximate lengths and diameters (in inches) o...
 4.3.109: A beaker of liquid at an initial temperature of 78C is placed in a ...
 4.3.110: Write a short paragraph explaining why the transformations of the d...
 4.3.111: In Exercises 111116, determine whether the statement is true or fal...
 4.3.112: In Exercises 111116, determine whether the statement is true or fal...
 4.3.113: In Exercises 111116, determine whether the statement is true or fal...
 4.3.114: In Exercises 111116, determine whether the statement is true or fal...
 4.3.115: In Exercises 111116, determine whether the statement is true or fal...
 4.3.116: In Exercises 111116, determine whether the statement is true or fal...
 4.3.117: Describe the error. ln( x2 x2 + 4) = ln x2 lnx2 + 4
 4.3.118: Consider the functions below. f(x) = ln x 2 , g(x) = ln x ln 2, h(x...
 4.3.119: For how many integers between 1 and 20 can the natural logarithms b...
 4.3.120: The figure shows the graphs of y = ln x, y = ln x2, y = ln 2x, and ...
 4.3.121: Does y1 = ln[x(x 2)] have the same domain as y2 = ln x + ln(x 2)? E...
 4.3.122: Prove that loga x logab x = 1 + loga 1 b .
 4.3.123: In Exercises 123 126, simplify the expression. 123. (64x3y4)3(8x3y2...
 4.3.124: In Exercises 123 126, simplify the expression. 123. (64x3y4)3(8x3y2...
 4.3.125: In Exercises 123 126, simplify the expression. 123. (64x3y4)3(8x3y2...
 4.3.126: In Exercises 123 126, simplify the expression. 123. (64x3y4)3(8x3y2...
Solutions for Chapter 4.3: Exponential and Logarithmic Functions
Full solutions for Algebra and Trigonometry: Real Mathematics, Real People  7th Edition
ISBN: 9781305071735
Solutions for Chapter 4.3: Exponential and Logarithmic Functions
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Chapter 4.3: Exponential and Logarithmic Functions includes 126 full stepbystep solutions. Since 126 problems in chapter 4.3: Exponential and Logarithmic Functions have been answered, more than 58487 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Algebra and Trigonometry: Real Mathematics, Real People, edition: 7. Algebra and Trigonometry: Real Mathematics, Real People was written by and is associated to the ISBN: 9781305071735.

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).

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

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

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.

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.

Iterative method.
A sequence of steps intended to approach the desired solution.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

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.

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

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

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

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!

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

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