 95.1: e 6
 95.2: e 3.4
 95.3: e 0.35
 95.4: ln 1.2
 95.5: ln 0.1
 95.6: ln 3.25
 95.7: e x = 4
 95.8: ln 1 = 0
 95.9: 2 e x  5 = 1
 95.10: 3 + e 2x = 8
 95.11: Find a formula for the height in terms of the outside air pressure.
 95.12: Use the formula you found in Exercise 11 to approximate the height ...
 95.13: e x > 30
 95.14: ln x < 6
 95.15: 2 ln 3x + 1 = 5
 95.16: ln x 2 = 9
 95.17: e 4
 95.18: e
 95.19: e 1.2
 95.20: e 0.5
 95.21: ln 3
 95.22: ln 10
 95.23: ln 5.42
 95.24: ln 0.03
 95.25: e x = 5
 95.26: e 2 = 6x
 95.27: ln e = 1
 95.28: ln 5.2 = x
 95.29: e x + 1 = 9
 95.30: e 1 = x 2
 95.31: ln _7 3 = 2x
 95.32: ln e x = 3
 95.33: 3 e x + 1 = 5
 95.34: 2 e x  1 = 0
 95.35: 3 e 4x + 11 = 2
 95.36: 8 + 3e 3x = 26
 95.37: 2 e x  3 = 1
 95.38: 2 e x + 3 = 0
 95.39: 2 + 3 e 3x = 7
 95.40: 1  _1 3 e 5x = 5
 95.41: According to this model, what will the worlds population be in 2015?
 95.42: Some experts have estimated that the worlds food supply can support...
 95.43: If you deposit $100 in an account paying 3.5% interest compounded c...
 95.44: Suppose you deposit A dollars in an account paying an interest rate...
 95.45: Explain why the equation you found in Exercise 44 might be referred...
 95.46: MAKE A CONJECTURE State a rule that could be used to approximate th...
 95.47: ln 2x = 4
 95.48: ln 3x = 5
 95.49: ln (x + 1) = 1
 95.50: ln (x  7) = 2
 95.51: e x < 4.5
 95.52: e x > 1.6
 95.53: e 5x 25
 95.54: e 2x 7
 95.55: If there are 156,000 people online, how many people will have recei...
 95.56: How much time will pass before half of the people will receive the ...
 95.57: ln x + ln 3x = 12
 95.58: ln 4x + ln x = 9
 95.59: ln ( x 2 + 12) = ln x + ln 8
 95.60: ln x + ln (x + 4) = ln 5
 95.61: OPEN ENDED Give an example of an exponential equation that requires...
 95.62: FIND THE ERROR Colby and Elsu are solving ln 4x = 5. Who is correct...
 95.63: CHALLENGE Determine whether the following statement is sometimes, a...
 95.64: Writing in Math Use the information about banking on page 536 to ex...
 95.65: ACT/SAT A recent study showed that the number of Australian homes w...
 95.66: REVIEW Which is the first incorrect step in simplifying log3 _ 3 48...
 95.67: log 4 68
 95.68: log 6 0.047
 95.69: log 50 23
 95.70: log 3 (a + 3) + log 3 (a  3) = log 3 16
 95.71: log 11 2 + 2 log 11 x = log 11 32
 95.72: mn = 4
 95.73: a b = c
 95.74: y = 7x
 95.75: BASKETBALL Alexis has never scored a 3point field goal, but she ha...
 95.76: 2 x = 10
 95.77: 5 x = 12
 95.78: 6 x = 13
 95.79: 2(1 + 0.1 ) x = 50
 95.80: 10(1 + 0.25 ) x = 200
 95.81: 400(1  0.2 ) x = 50
Solutions for Chapter 95: Base e and Natural Logarithms
Full solutions for Algebra 2, Student Edition (MERRILL ALGEBRA 2)  1st Edition
ISBN: 9780078738302
Solutions for Chapter 95: Base e and Natural Logarithms
Get Full SolutionsThis textbook survival guide was created for the textbook: Algebra 2, Student Edition (MERRILL ALGEBRA 2), edition: 1. This expansive textbook survival guide covers the following chapters and their solutions. Algebra 2, Student Edition (MERRILL ALGEBRA 2) was written by and is associated to the ISBN: 9780078738302. Chapter 95: Base e and Natural Logarithms includes 81 full stepbystep solutions. Since 81 problems in chapter 95: Base e and Natural Logarithms have been answered, more than 52223 students have viewed full stepbystep solutions from this chapter.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

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.

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).

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

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

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.

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.

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].

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

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!).

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

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

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

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·

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.