 9.5.1: Use a calculator to evaluate each expression to four decimal places.
 9.5.2: Use a calculator to evaluate each expression to four decimal places.
 9.5.3: Use a calculator to evaluate each expression to four decimal places.
 9.5.4: Use a calculator to evaluate each expression to four decimal places.
 9.5.5: Use a calculator to evaluate each expression to four decimal places.
 9.5.6: Use a calculator to evaluate each expression to four decimal places.
 9.5.7: Write an equivalent exponential or logarithmic equation.
 9.5.8: Write an equivalent exponential or logarithmic equation.
 9.5.9: Solve each equation. Round to the nearest tenthousandth.
 9.5.10: Solve each equation. Round to the nearest tenthousandth.
 9.5.11: Find a formula for the height in terms of the outside air pressure.
 9.5.12: Use the formula you found in Exercise 11 to approximate the height ...
 9.5.13: Solve each equation or inequality. Round to the nearest tenthousan...
 9.5.14: Solve each equation or inequality. Round to the nearest tenthousan...
 9.5.15: Solve each equation or inequality. Round to the nearest tenthousan...
 9.5.16: Solve each equation or inequality. Round to the nearest tenthousan...
 9.5.17: Use a calculator to evaluate each expression to four decimal places
 9.5.18: Use a calculator to evaluate each expression to four decimal places
 9.5.19: Use a calculator to evaluate each expression to four decimal places
 9.5.20: Use a calculator to evaluate each expression to four decimal places
 9.5.21: Use a calculator to evaluate each expression to four decimal places
 9.5.22: Use a calculator to evaluate each expression to four decimal places
 9.5.23: Use a calculator to evaluate each expression to four decimal places
 9.5.24: Use a calculator to evaluate each expression to four decimal places
 9.5.25: Write an equivalent exponential or logarithmic equation.
 9.5.26: Write an equivalent exponential or logarithmic equation.
 9.5.27: Write an equivalent exponential or logarithmic equation.
 9.5.28: Write an equivalent exponential or logarithmic equation.
 9.5.29: Write an equivalent exponential or logarithmic equation.
 9.5.30: Write an equivalent exponential or logarithmic equation.
 9.5.31: Write an equivalent exponential or logarithmic equation.
 9.5.32: Write an equivalent exponential or logarithmic equation.
 9.5.33: Solve each equation. Round to the nearest tenthousandth.
 9.5.34: Solve each equation. Round to the nearest tenthousandth.
 9.5.35: Solve each equation. Round to the nearest tenthousandth.
 9.5.36: Solve each equation. Round to the nearest tenthousandth.
 9.5.37: Solve each equation. Round to the nearest tenthousandth.
 9.5.38: Solve each equation. Round to the nearest tenthousandth.
 9.5.39: Solve each equation. Round to the nearest tenthousandth.
 9.5.40: Solve each equation. Round to the nearest tenthousandth.
 9.5.41: According to this model, what will the worlds population be in 2015?
 9.5.42: Some experts have estimated that the worlds food supply can support...
 9.5.43: If you deposit $100 in an account paying 3.5% interest compounded c...
 9.5.44: Suppose you deposit A dollars in an account paying an interest rate...
 9.5.45: Explain why the equation you found in Exercise 44 might be referred...
 9.5.46: State a rule that could be used to approximate the amount of time t...
 9.5.47: Solve each equation or inequality. Round to the nearest tenthousan...
 9.5.48: Solve each equation or inequality. Round to the nearest tenthousan...
 9.5.49: Solve each equation or inequality. Round to the nearest tenthousan...
 9.5.50: Solve each equation or inequality. Round to the nearest tenthousan...
 9.5.51: Solve each equation or inequality. Round to the nearest tenthousan...
 9.5.52: Solve each equation or inequality. Round to the nearest tenthousan...
 9.5.53: Solve each equation or inequality. Round to the nearest tenthousan...
 9.5.54: Solve each equation or inequality. Round to the nearest tenthousan...
 9.5.55: If there are 156,000 people online, how many people will have recei...
 9.5.56: How much time will pass before half of the people will receive the ...
 9.5.57: Solve each equation. Round to the nearest tenthousandth.
 9.5.58: Solve each equation. Round to the nearest tenthousandth.
 9.5.59: Solve each equation. Round to the nearest tenthousandth.
 9.5.60: Solve each equation. Round to the nearest tenthousandth.
 9.5.61: Give an example of an exponential equation that requires using natu...
 9.5.62: Colby and Elsu are solving ln 4x = 5. Who is correct? Explain your ...
 9.5.63: Determine whether the following statement is sometimes, always, or ...
 9.5.64: Use the information about banking on page 536 to explain how the na...
 9.5.65: A recent study showed that the number of Australian homes with a co...
 9.5.66: Which is the first incorrect step in simplifying log3 _ 3 48 ? Step...
 9.5.67: Express each logarithm in terms of common logarithms. Then approxim...
 9.5.68: Express each logarithm in terms of common logarithms. Then approxim...
 9.5.69: Express each logarithm in terms of common logarithms. Then approxim...
 9.5.70: Solve each equation. Check your solutions
 9.5.71: Solve each equation. Check your solutions
 9.5.72: State whether each equation represents a direct, joint, or inverse ...
 9.5.73: State whether each equation represents a direct, joint, or inverse ...
 9.5.74: State whether each equation represents a direct, joint, or inverse ...
 9.5.75: Alexis has never scored a 3point field goal, but she has scored a ...
 9.5.76: Solve each equation. Round to the nearest hundredth.
 9.5.77: Solve each equation. Round to the nearest hundredth.
 9.5.78: Solve each equation. Round to the nearest hundredth.
 9.5.79: Solve each equation. Round to the nearest hundredth.
 9.5.80: Solve each equation. Round to the nearest hundredth.
 9.5.81: Solve each equation. Round to the nearest hundredth.
Solutions for Chapter 9.5: Base e and Natural Logarithms
Full solutions for California Algebra 2: Concepts, Skills, and Problem Solving  1st Edition
ISBN: 9780078778568
Solutions for Chapter 9.5: Base e and Natural Logarithms
Get Full SolutionsThis textbook survival guide was created for the textbook: California Algebra 2: Concepts, Skills, and Problem Solving, edition: 1. Since 81 problems in chapter 9.5: Base e and Natural Logarithms have been answered, more than 44549 students have viewed full stepbystep solutions from this chapter. Chapter 9.5: Base e and Natural Logarithms includes 81 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. California Algebra 2: Concepts, Skills, and Problem Solving was written by and is associated to the ISBN: 9780078778568.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

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.

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

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.

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

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

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.

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!).

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

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

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

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.

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

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.

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

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.

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

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