 5.3.1: To evaluate a logarithm to any base, you can use the ________ formula.
 5.3.2: The changeofbase formula for base is given by ________.
 5.3.3: You can consider to be a constant multiple of the constant multipli...
 5.3.4: loga uv loga u loga v
 5.3.5: ln un n ln u
 5.3.6: loga u v loga u loga v
 5.3.7:
 5.3.8:
 5.3.9:
 5.3.10: log1 3 log x
 5.3.11: logx 3 10
 5.3.12: logx 3 4
 5.3.13: log x 2.6 x
 5.3.14: log7.1 log x
 5.3.15: log3 7
 5.3.16: log7 log3 7 4
 5.3.17: log1 2 4
 5.3.18: log1 4 log1 2 4 5
 5.3.19: log9 0.1
 5.3.20: log20 log9 0.1 0.25
 5.3.21: log15 1250
 5.3.22: log3 log15 1250 0.015
 5.3.23: log 4 8
 5.3.24: log2 42 34
 5.3.25: log5 1 250
 5.3.26: log 9 300
 5.3.27: ln 5e6
 5.3.28: ln 6 e2
 5.3.29: log3 9
 5.3.30: log5 1 125
 5.3.31: log2 6 4 8
 5.3.32: log6 3 log2 6
 5.3.33: log4 162
 5.3.34: log3 813 l
 5.3.35: log 27 2 2 lo
 5.3.36: log3 log 27 2
 5.3.37: ln e4.5
 5.3.38: 3 ln e4
 5.3.39: ln 1 e
 5.3.40: ln 4 e3 l
 5.3.41: ln e2 ln e5
 5.3.42: 2 ln e6 ln e5 l
 5.3.43: log5 75 log5 3 3
 5.3.44: log4 2 log4
 5.3.45: ln 4x
 5.3.46: log3 ln 4x 10z
 5.3.47: log8 x 4
 5.3.48: log10 y 2
 5.3.49: log5 5 x
 5.3.50: log6 1 z3
 5.3.51: ln z t
 5.3.52: ln 3
 5.3.53: ln xyz y 2
 5.3.54: log 4x2 ln xyz y
 5.3.55: ln z z 1 , x > 1 2, z > 1
 5.3.56: ln x 2 1 x3 ln z z 1 , x > 1 2,
 5.3.57: log2 a 1 9 , a > 1
 5.3.58: ln 6 x 2 1 l
 5.3.59: ln 3 x y
 5.3.60: ln x 2 y3 l
 5.3.61: ln x2 y z
 5.3.62: log2 x4 y z3 l
 5.3.63: log5 x 2 y 2z3
 5.3.64: log10 xy4 z5
 5.3.65: ln x 2 4 x3 x2 3
 5.3.66: lnx 2 ln x 2
 5.3.67: ln 2 ln x
 5.3.68: ln y ln t
 5.3.69: log t 4 z log4 y l
 5.3.70: log5 8 log5 log t 4
 5.3.71: 2 log2 x 4 log2 y
 5.3.72: 2 3 log7 z 2
 5.3.73: 4 log3 5x
 5.3.74: 4 log6 2x
 5.3.75: log x 2 log x 1
 5.3.76: 2 ln 8 5 ln z 4
 5.3.77: log x 2 log y 3 log z
 5.3.78: 3 log3 x 4 log3 y 4 log3 z
 5.3.79: ln x ln x 1 ln x 1 3
 5.3.80: 4 ln z ln z 5 2 ln z 5 l
 5.3.81: 1 3 2 ln x 3 ln x ln x2 1 4
 5.3.82: 2 3 ln x ln x 1 ln x 1 1
 5.3.83: 3 log8 y 2 log8 y 4 log8 y 1 2
 5.3.84: 2 log4 x 1 2 log4 x 1 6 log4 x
 5.3.85: log2 32 log2 4 ,
 5.3.86: log770, log7 35, 1 2 log7 10 l
 5.3.87: Use the properties of logarithms to write the formula in simpler fo...
 5.3.88: Find the difference in loudness between an average office with an i...
 5.3.89: Find the difference in loudness between a vacuum cleaner with an in...
 5.3.90: You and your roommate are playing your stereos at the same time and...
 5.3.91: x 12 3 4 5 6 y 1 1.189 1.316 1.414 1.495 1.565
 5.3.92: x 12 3 4 5 6 y 1 1.587 2.080 2.520 2.924 3.302
 5.3.93: x 1234 5 6 y 2.5 2.102 1.9 1.768 1.672 1.597
 5.3.94: x 12 34 5 6 y 0.5 2.828 7.794 16 27.951 44.091
 5.3.95: GALLOPING SPEEDS OF ANIMALS Fourlegged animals run with two differ...
 5.3.96: NAIL LENGTH The approximate lengths and diameters (in inches) of co...
 5.3.97: COMPARING MODELS A cup of water at an initial temperature of is pla...
 5.3.98: PROOF Prove that
 5.3.99: PROOF Prove that
 5.3.100: CAPSTONE A classmate claims that the following are true. (a) (b) (c...
 5.3.101: f 0 0
 5.3.102: f ax f a f x, a > 0, x > 0
 5.3.103: f x 2 f x f 2, x > 2 f
 5.3.104: f x 1 2 f x
 5.3.105: If then
 5.3.106: If then
 5.3.107: f x log2 x
 5.3.108: f x log4 x
 5.3.109: f x log1 2 x
 5.3.110: f x log1 4 x
 5.3.111: f x log11.8 x
 5.3.112: f x log12.4 x
 5.3.113: THINK ABOUT IT Consider the functions below. Which two functions sh...
 5.3.114: GRAPHICAL ANALYSIS Use a graphing utility to graph the functions gi...
 5.3.115: THINK ABOUT IT For how many integers between 1 and 20 can the natur...
Solutions for Chapter 5.3: Properties of Logarithms
Full solutions for Algebra and Trigonometry  8th Edition
ISBN: 9781439048474
Solutions for Chapter 5.3: Properties of Logarithms
Get Full SolutionsSince 115 problems in chapter 5.3: Properties of Logarithms have been answered, more than 47332 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Algebra and Trigonometry, edition: 8. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 5.3: Properties of Logarithms includes 115 full stepbystep solutions. Algebra and Trigonometry was written by and is associated to the ISBN: 9781439048474.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

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

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

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!).

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

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.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

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

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

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

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

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!

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.

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).