 B.4.B.4.4: Find the following logarithms. log 378
 B.4.B.4.5: Find the following logarithms. log 37.8
 B.4.B.4.6: Find the following logarithms. log 3,780
 B.4.B.4.7: Find the following logarithms. log 0.4260
 B.4.B.4.8: Find the following logarithms. log 0.0378
 B.4.B.4.9: Find the following logarithms. log 37,800
 B.4.B.4.10: Find the following logarithms. log 600
 B.4.B.4.11: Find the following logarithms. log 10,200
 B.4.B.4.12: Find the following logarithms. log 0.00971
 B.4.B.4.13: Find the following logarithms. log 0.0314
 B.4.B.4.14: Find the following logarithms. log 0.00052
 B.4.B.4.15: Find the following logarithms. log 0.399
 B.4.B.4.16: Find x in the following equations. log x 2.8802
 B.4.B.4.17: Find x in the following equations. log x 4.8802
 B.4.B.4.18: Find x in the following equations. logx2.1198
 B.4.B.4.19: Find x in the following equations. logx 3.1198
 B.4.B.4.20: Find x in the following equations. logx 5.3497
 B.4.B.4.21: Find x in the following equations. logx1.5670
 B.4.B.4.22: Find x without using a calculator. log x = 10
 B.4.B.4.23: Find x without using a calculator. log x =1
 B.4.B.4.24: Find x without using a calculator. log x = 10
 B.4.B.4.25: Find x without using a calculator. log x = 1
 B.4.B.4.26: Find x without using a calculator. log x = 20
 B.4.B.4.27: Find x without using a calculator. log x = 20
 B.4.B.4.28: Find x without using a calculator. log x = 2
 B.4.B.4.29: Find x without using a calculator. log x = 4
 B.4.B.4.30: Find x without using a calculator. log x = log2 8
 B.4.B.4.31: Find x without using a calculator. log X log3 9
 B.4.B.4.32: Simplify each of the following expressions. In e
 B.4.B.4.33: Simplify each of the following expressions. In 1
 B.4.B.4.34: Simplify each of the following expressions. In e 3
 B.4.B.4.35: Simplify each of the following expressions. In ~
 B.4.B.4.36: Use the properties of logarithms to expand each of the following ex...
 B.4.B.4.37: Use the properties of logarithms to expand each of the following ex...
 B.4.B.4.38: Use the properties of logarithms to expand each of the following ex...
 B.4.B.4.39: Use the properties of logarithms to expand each of the following ex...
 B.4.B.4.40: Find a decimal approximation to each of the following natural logar...
 B.4.B.4.41: Find a decimal approximation to each of the following natural logar...
 B.4.B.4.42: Find a decimal approximation to each of the following natural logar...
 B.4.B.4.43: Find a decimal approximation to each of the following natural logar...
 B.4.B.4.44: Find a decimal approximation to each of the following natural logar...
 B.4.B.4.45: Find a decimal approximation to each of the following natural logar...
 B.4.B.4.46: Find a decimal approximation to each of the following natural logar...
 B.4.B.4.47: Find a decimal approximation to each of the following natural logar...
 B.4.B.4.48: IfIn 2 0.6931, In 3 1.0986, and In 5 = 1.6094, find each of the fol...
 B.4.B.4.49: IfIn 2 0.6931, In 3 1.0986, and In 5 = 1.6094, find each of the fol...
 B.4.B.4.50: IfIn 2 0.6931, In 3 1.0986, and In 5 = 1.6094, find each of the fol...
 B.4.B.4.51: IfIn 2 0.6931, In 3 1.0986, and In 5 = 1.6094, find each of the fol...
 B.4.B.4.52: IfIn 2 0.6931, In 3 1.0986, and In 5 = 1.6094, find each of the fol...
 B.4.B.4.53: IfIn 2 0.6931, In 3 1.0986, and In 5 = 1.6094, find each of the fol...
 B.4.B.4.54: IfIn 2 0.6931, In 3 1.0986, and In 5 = 1.6094, find each of the fol...
 B.4.B.4.55: IfIn 2 0.6931, In 3 1.0986, and In 5 = 1.6094, find each of the fol...
 B.4.B.4.56: Measuring Acidity Previously we indicated that the pH of a solution...
 B.4.B.4.57: Measuring Acidity Previously we indicated that the pH of a solution...
 B.4.B.4.58: Measuring Acidity Previously we indicated that the pH of a solution...
 B.4.B.4.59: Measuring Acidity Previously we indicated that the pH of a solution...
 B.4.B.4.60: The Richter Scale Find the relative size T of the shock wave of ear...
 B.4.B.4.61: The Richter Scale Find the relative size T of the shock wave of ear...
 B.4.B.4.62: The Richter Scale Find the relative size T of the shock wave of ear...
 B.4.B.4.63: The Richter Scale Find the relative size T of the shock wave of ear...
 B.4.B.4.64: Shock Wave How much larger is the shock wave of an earthquake that ...
 B.4.B.4.65: Shock Wave How much larger is the shock wave of an earthquake that ...
 B.4.B.4.66: Earthquake Table I gives a partial listing of earthquakes that were...
 B.4.B.4.67: Earthquake On January 6, 200 I, an earthquake with a magnitude of 7...
 B.4.B.4.68: Depreciation The annual rate of depreciation r on a car that is pur...
 B.4.B.4.69: Depreciation The annual rate of depreciation r on a car that is pur...
 B.4.B.4.70: Two cars depreciate in value according to the following depreciatio...
 B.4.B.4.71: Two cars depreciate in value according to the following depreciatio...
 B.4.B.4.72: Getting Close to e Use a calculator to complete the following table...
 B.4.B.4.73: Getting Close to e Use a calculator to complete the following table...
Solutions for Chapter B.4: Exponential and Logarithmic Functions
Full solutions for Trigonometry
ISBN: 9780495108351
Solutions for Chapter B.4: Exponential and Logarithmic Functions
Get Full SolutionsTrigonometry was written by and is associated to the ISBN: 9780495108351. This textbook survival guide was created for the textbook: Trigonometry, edition: . Chapter B.4: Exponential and Logarithmic Functions includes 70 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 70 problems in chapter B.4: Exponential and Logarithmic Functions have been answered, more than 31791 students have viewed full stepbystep solutions from this chapter.

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.

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

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

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

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.

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

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.

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.

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.

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

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.

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.