Test 4

If an exponential equation can be written in the form bu=bv, then which of the following methods may be used to solve the​ equation? A. Subtracting bv from both sides. B. Bringing down the exponent on each side. C. Dividing both sides by b. D. Relating the bases by setting

D. Relating the bases by setting

Which of the following statements is not true about a rational function of the form f(x)=g(x)h(x) where g and h are polynomial​ functions? A. If the degree of g is m and the degree of h is n such that m=n​, then f will have a horizontal asymptote with equation y=anbm where an is the leading coefficient of g and bm is the leading coefficient of h. B. The graph of a rational function will never intersect a vertical asymptote. C. A rational function may have many vertical asymptotes. D. A rational function may have many horizontal asymptotes.

D. A rational function may have many horizontal asymptotes.

In solving a logarithmic equation of the form logb(x+a)+logb(x+c)=d, why is it essential to check the​ solution(s) to the resulting quadratic equation for potential extraneous solutions to the original logarithmic​ equation? A. Any solution must be included in either the interval x≤−c or x≤−a. B. Equations of this form can never have more than one solution. C. Negative numbers cannot be solutions to logarithmic equations. D. All solutions must satisfy x+a>0 and x+c>0.

D. All solutions must satisfy x+a>0 and x+c>0.

Which of the following is​ true? A. lne^8=8 B. (log4x)^2=2log4x C. ln6x2=2ln6x D. log5(2x3)=3log5(2x)

A. lne^8=8

Which of the following is not​ true? A. lnx+ln2x=ln3x B. ln5x−ln1=ln5x C. lnx−1x2+4=ln(x−1)−ln(x2+4) D. 1/2log(x-1)-3logz+log5=log5squarerootx-1/z^3

A. lnx+ln2x=ln3x

Which of the following statements is not true about a rational function of the form f(x)=g(x)h(x) where g and h are polynomial​ functions? A. If​ f(x) has a y​-​intercept, it can be found by evaluating​ f(0) provided that​ f(0) is defined. B. The domain of​ f(x) consists of all values of x such that g(x)≠0 and h(x)≠0. C. The function f(x)=g(x)h(x) can have an x​-intercept at x=0. D. If​ f(x) has any​ x-intercepts, they can be found by solving the equation g(x)=0 provided that​ g(x) and​ h(x) have no common factors.

B. The domain of​ f(x) consists of all values of x such that g(x)≠0 and h(x)≠0.

The domain of f(x)=logb[g(x)] can be determined by finding the solution to which​ inequality? A. g(x)≤0 B. g(x)<0 C. g(x)≥0 D. g(x)>0

D. g(x)>0

Which of the following statements is not true for the graph of y=logbx for b>1? A. The graph of y=logbx is decreasing on the interval ​(0,∞​). B. The graph of y=logbx contains the point​ (1,0). C. The graph of y=logbx contains the point​ (b,1). D. The line x=0 is a vertical asymptote.

A. The graph of y=logbx is decreasing on the interval ​(0,∞​).

For x>0, b>0 and b≠1, if y=logbx, then which of the following is​ true? A. x=by B. y=bx C. x=yb D. y=xb

A. x=by

In solving an exponential​ equation, which of the following is not a sound technique to​ use? A. If the exponential equation has the form abx=c, first divide both sides by the constant a. B. If the exponential equation cannot be written in the form bu=bv, ​"take the log of both​ sides" and then​ "bring down any​ exponents." C. If the exponential equation has the form abx=c, first​ "take the log of both​ sides" and then​ "bring down any​ exponents." D. ​First, try writing the exponential equation in the form bu=bv and then solving u=v

C. If the exponential equation has the form abx=c, first​ "take the log of both​ sides" and then​ "bring down any​ exponents."

Why is the logarithmic property of​ equality, which says that​ "if logbu=logbv, then u=v" ​true? A. It is true because the logarithmic function always intersects the​ x-axis at the point (1,0). B. It is true because the logarithmic function has a vertical asymptote. C. It is true because the logarithmic function is​ one-to-one. D. It is true because the logarithmic function is an increasing function.

C. It is true because the logarithmic function is​ one-to-one.

Which of the following statements is not true for the graph of f(x)=ex​? A. The graph of f(x)=ex intersects the y​-axis at​ (0,1). B. The graph of f(x)=ex approaches 0 as x approaches negative infinity. C. The graph of f(x)=ex lies between the graphs of y=3x and y=4x. D. The line y=0 is a horizontal asymptote.

C. The graph of f(x)=ex lies between the graphs of y=3x and y=4x.

Which of the following statements about the number e is not​ true? A. The number e is called the natural base. B. The number e is defined as the value of the expression 1+1nn as n approaches infinity. C. The number e is a rational number. D. The number e is an irrational number.

C. The number e is a rational number.

In the definition of the exponential function f(x)=bx, what are the​ stipulation(s) for the base​ b? A. The base b cannot be a fraction. B. The base b must be greater than or equal to zero. C. The base b must be greater than one. D. The base b must be greater than zero and not equal to 1.

D. The base b must be greater than zero and not equal to 1.

Which of the following statements is not true for the graph of f(x)=bx, where 0<b<1? A. The line y=0 is a horizontal asymptote. B. The graph intersects the​ y-axis at​ (0,1). C. The graph of f(x)=bx approaches 0 as x approaches infinity. D. The graph of f(x)=bx approaches 0 as x approaches negative infinity.

D. The graph of f(x)=bx approaches 0 as x approaches negative infinity.

Logarithms are studied for which of the following​ reasons? A. To validate the logarithmic properties. B. To be able to solve complex logarithmic equations. C. To help in solving exponential equations when relating the bases cannot be used. D. To make​ student's lives miserable.

C. To help in solving exponential equations when relating the bases cannot be used.

To what is the expression logbbx​, for b>0 and b≠1​, ​equal? A. e B. b C. 1 D. x

D. x

Which of the following statements is true about horizontal asymptotes of a rational function of the form f(x)=g(x)h(x) where g and h are polynomial​ functions? A. The function f(x)=g(x)h(x) will have a horizontal asymptote only if the degree of g is less than or equal to the degree of h. B. The function f(x)=g(x)h(x) will have a horizontal asymptote only if the degree of g is less than the degree of h. C. Every rational function has a horizontal asymptote. D. The function f(x)=g(x)h(x) will have a horizontal asymptote only if the degree of g is equal to the degree of h.

A. The function f(x)=g(x)h(x) will have a horizontal asymptote only if the degree of g is less than or equal to the degree of h. Your answer is correct.

Which of the following statements is true about vertical asymptotes of a rational function of the form f(x)=g(x)h(x) where g and h are polynomial​ functions? A. If a is a constant and h(a)=0​, then​ f(x) must have a vertical asymptote at x=a. B. In order to correctly determine the vertical​ asymptotes, it is essential to cancel any common factors of g and h. C. Every rational function has at least one vertical asymptote. D. To determine the behavior of a rational function near the vertical asymptote from the left of the​ asymptote, the sign of the function must be determined using any test value to the left of the asymptote.

B. In order to correctly determine the vertical​ asymptotes, it is essential to cancel any common factors of g and h.

In solving the equation ln(x−1)=2, what is the first​ step? A. Add the constant to both sides. B. Rewrite as an exponential equation. C. Take the log of both sides. D. Substitute e for

B. Rewrite as an exponential equation.

Which of the following statements is true about a rational function of the form f(x)=g(x)h(x) where g and h are polynomial​ functions? A. The rational function f(x)=g(x)h(x) will have a removable discontinuity at x=a if g(a)=0. B. If the rational function has a removable​ discontinuity, then it cannot have a horizontal asymptote. C. If the rational function has a removable​ discontinuity, then it cannot have a vertical asymptote. D. The rational function f(x)=g(x)h(x) will have a removable discontinuity only if​ g(x) and​ h(x) share a common factor.

D. The rational function f(x)=g(x)h(x) will have a removable discontinuity only if​ g(x) and​ h(x) share a common factor.

If a and b are positive real numbers such that a≠1​, b≠1, and u is any positive real​ number, then the logarithmic expression logbu is equivalent to which of the​ following? A. log^ab/log^au B. log^ub/log^ua C. log^ua/log^ub D. log^au/log^ab

D. log^au/log^ab

If b>0, b≠1, u and v represent positive​ numbers, and r is any real​ number, which of the following statements is not a property of​ logarithms? A. logb1=0 B. logbu/v=logbu−logbv C. logbu^r=rlogbu D. logb(u+v)=logbu+logbv

D. logb(u+v)=logbu+logbv