Math Exam 4

a. loga u/loga b

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 logb u is equivalent to which of the following? a. loga u/loga b b. logu a/logu b c. logu b/logu a d. loga b/loga u

a. Relating the bases by setting u=v.

If an exponential equation can be written in the form b^u=b^v, then which of the following methods may be used to solve the equation? a. Relating the bases by setting u=v. b. Bringing down the exponent on each side. c. Dividing both sides by b. d. Subtracting b^v from both sides.

b. If the exponential equation has the form ab^x=c, first "take the log of both sides" and then "bring down any exponents".

In solving an exponential equation, which of the following is not a sound technique to use? a. First, try writing the exponential equation in the form b^u=b^v and then u=v. b. If the exponential equation has the form ab^x=c, first "take the log of both sides" and then "bring down any exponents". c. If the exponential equation cannot be written in the form b^u=b^v, "take the log of both sides" and then "bring down any exponents". d. If the exponential equation has the form ab^x=c, first divide both sides by the constant a.

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

In solving the 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. Equations of this form can never have more than one solution. b. All solutions must satisfy x+a>0 and x+c>0. c. Any solution must be included in either the interval x≤-c or x≤-a. d. Negative numbers cannot be solutions to logarithmic equations.

a. ln e^8=8

Which of the following is true? a. ln e^8=8 b. log5 (2x^3)=3log5 (2x) c. ln 6x^2=2ln 6x d. (log4 x)^2=2log4 x

c. 2.7183

Which of the following numbers is best approximation of the number e to 4 decimal places? a. 3.1415 b. 7.1461 c. 2.7183 d. 1.6278

d. The number e is a rational number.

Which of the following statements about the number e is not true? a. The number e is an irrational number. b. The number e is defined as the value of the expression (1+1/n)^n as n approaches infinity. c. The number e is called the natural base. d. The number e is a rational number.

b. logb(u+v)=logb u+logb v

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. logb u/v=logb u-logb v b. logb(u+v)=logb u+logb v c. logb u^r=rlogb u d. logb 1=0

d. This expression represents the number 7.

Which of the following is not true about the expression log7 49? a. This expression is called a logarithms expression. b. This expression represents the power that 7 can be raised to in order to get 49. c. This expression represents the number 2. d. This expression represents the number 7.

d. The base b must be greater than zero but not equal to 1.

In the definition of the exponential function f(x)=b^x, what are the stipulations(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 but not equal to 1.

d. The line y=2 is a horizontal asymptote.

Which of the following statements is true for the graph of f(x)=-e^x +2? a. The x-axis is a horizontal asymptote. b. The y-axis is a vertical asymptote. c. The line x=2 is a vertical asymptote. d. The line y=2 is a horizontal asymptote.

d. It is true because the logarithmic function is one-to-one.

Why is the logarithmic property of equality, which says that "if logb u=logb v, 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 is an increasing function. c. It is true because the logarithmic function has a vertical asymptote. d. It is true because the logarithmic function is one-to-one.

b. g(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

b. ln x+ln 2x=ln 3x

Which of the following is not true? a. ln 5x-ln 1=ln 5x b. ln x+ln 2x=ln 3x c. 1/2 log(x-1)-3log z+log (5 square root (x-1))/z^3 d. ln (x-1)/(x^2+4) =ln (x-1)-ln (x^2+4)

b. The graph of f(x)=e^x lies between the graphs of y=3^x and y=4^x.

Which of the following statements is not true for the graph of f(x)=e^x? a. The graph of f(x)=e^x approaches 0 as x approaches negative infinity. b. The graph of f(x)=e^x lies between the graphs of y=3^x and y=4^x. c. The line y=0 is a horizontal asymptote. d. The graph of f(x)=e^x intersects the y-axis at (0,1).

b. A rational function may have many horizontal asymptotes.

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. The graph of a rational function will never intersect a vertical asymptote. b. A rational function may have many horizontal asymptotes. c. A rational function may have many vertical asymptotes. d. 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 the equation y=an/bm where an is the leading coefficient of g and bm is the leading coefficient of h.

c. Rewrite as an exponential equation.

In solving the equation ln (x-1)=2, what is the first step? a. Take the log of both sides. b. Substitute e for ln x. c. Rewrite as an exponential equation. d. Add the constant to both sides.

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

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

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

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. The domain of f(x) consists of all values of x such that g(x)≠0 and h(x)≠0. b. 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. c. If f(x) has a y-intercept, it can be found by evaluating f(0) provided that f(0) is defined. d. The function f(x)=g(x)/h(x) can have an x-intercept at x=0.

c. The graph of f(x)=b^x approaches 0 as x approaches negative infinity.

Which of the following statements is not true for the graph f(x)=b^x, 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)=b^x approaches 0 as x approaches negative infinity. d. The graph of f(x)=b^x approaches 0 as x approaches infinity.

d. The graph of y=logb x is decreasing on the interval (0,∞).

Which of the following statements is not true for the graph of y=logb x for b>1? a. The graph of y=logb x contains the point (1,0). b. The graph of y=logb x contains the point (b,1). c. The line x=0 is a vertical asymptote. d. The graph of y=logb x is decreasing on the interval (0,∞).

b. 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.

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 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 or equal to the degree of h. c. 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. d. Every rational function has a horizontal asymptote.

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

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. Every rational function has at least one vertical asymptote. b. 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. c. If a is a constant and h(a)=0, then f(x) must have a vertical asymptote at x=a. d. In order to correctly determine the vertical asymptotes, it is essential to cancel any common factors of g and h.

c. All boundary points of a rational inequality that are found by determining the values for which the denominator is equal to zero should always be represented by plotting an open circle on a number line.

Which of the following statements is true? a. All boundary points of a rational inequality that are found by determining the values for which the numerator is equal to zero should always be represented by plotting an open circle on a number line. b. All boundary points of a rational inequality should always be represented by plotting a closed circle on a number line. c. All boundary points of a rational inequality that are found by determining the values for which the denominator is equal to zero should always be represented by plotting an open circle on a number line. d. All boundary points of a rational inequality should always be represented by plotting an open circle on a number line.

a. If xln 4/7=-ln 7, then x=-0.2005.

In solving the exponential equation 4^x=7^(x-1), which of the following is not true? a. If xln 4/7=-ln 7, then x=-0.2005. b. If xln 4-xln 7=-ln 7, then x=(-ln 7)/(ln 4/7). c. If 4^x=7^(x-1), then xln 4=(x-1)ln 7. d. If xln 4=(x-1)ln 7, then xln 4-xln 7=-ln7.

b. x=b^y

For x>0, b>0 and b≠1, if y=logb x, then which of the following is true? a. y=x^b b. x=b^y c. y=b^x d. x=y^b

c. x

To what is the expression logb b^x, for b>0 and b≠1, equal? a. 1 b. e c. x d. b