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All Textbook Solutions for College Algebra
For the following exercises, use the Rational Zero Theorem to help you solve the polynomial equation. 28. 4x4+8x3+19x2+32x+12=0For the following exercises, use Descartes’ Rule of Signs to find the possible number of positive and negative solutions. x33x22x+4=0For the following exercises, use Descartes’ Rule of Signs to find the possible number of positive and negative solutions. 2x4x3+4x25x+1=0For the following exercises. find the intercepts and the vertical and horizontal asymptotes, and then use them to sketcha graph of the function. 31. f(x)=x+2x5For the following exercises. find the intercepts and the vertical and horizontal asymptotes, and then use them to sketcha graph of the function. 32. f(x)=x2+1x24For the following exercises, find the intercepts and the vertical and horizontal asymptotes, and then use them to sketcha graph of the function. 33. f(x)=3x227x2+x2For the following exercises. find the intercepts and the vertical and horizontal asymptotes, and then use them to sketcha graph of the function. 34. f(x)=x+2x29For the following exercises, find the slant asymptote. f(x)=x21x+2For the following exercises, find the slant asymptote. f(x)=2x3x2+4x2+1For the following exercises, find the inverse of the function with the domain given. 37. f(x)=(x2)2,x2For the following exercises, find the inverse of the function with the domain given. 38. f(x)=(x+4)23,x4For the following exercises, find the inverse of the function with the domain given. 39. f(x)=x2+6x2,x3For the following exercises, find the inverse of the function with the domain given. 40. f(x)=2x33For the following exercises, find the inverse of the function with the domain given. 41. f(x)=4x+53For the following exercises, find the inverse of the function with the domain given. 42. f(x)=x32x+1For the following exercises, find the unknown value. 43. y varies directly as the square of x. If when x=3,y=36,findyifx=4.For the following exercises, find the unknown value. 44. y varies inversely as the square root ofx. If when x=25,y=2,findyifx=4.For the following exercises, find the unknown value. 45. y varies jointly as the cube of x and as z. If when x=1 and z=2,y=6,findyifx=2 and z=3.For the following exercises, find the unknown value. 46. y varies jointly as x and the square ofz and inverselyasthe cubeofw. Ifwhen x=3,z=4, and w=2,y=48,findy if x=4,z=5, and w=3.For the following exercises, solve the application problem. 47. The weight of an object above the surface of theearth varies inversely with the distance from thecenter of the earth. If a person weighs 150 poundswhen he is on the surface of the earth (3,960 milesfrom center), find the weight of the person if he is 26miles above the surface.For the following exercises, solve the application problem. 48.The volume V of an ideal gas varies directly with thetemperature T and inversely with the pressure P.A cylinder contains oxygen at a temperature of310 degrees K and a pressure of 18 atmospheres in avolume of 120 liters. Find the pressure if the volumeis decreased to 100 liters and the temperature isincreased to 320 degrees K.Give the degree and leading coefficient of the following polynomial function. f(x)=x3(36x22x2)Determine the and behavior of the polynomial function. f(x)=8x33x2+2x4Determine the and behavior of the polynomial function. f(x)=2x2(43x5x2)Write the quadratic function in standard form. Determine the vertex and axes intercepts and graph the function. f(x)=x2+2x8Given information about the graph of a quadratic function, find its equation. Vertex (2,0) and point on graph (4,12).Solve the following application problem. A rectangular field is to be enclosed by fencing. In addition to the enclosing fence, another fence is to divide the field into two parts, running parallel to two sides. If 1,200 feet of fencing is available, find the maximum area that can be enclosed.Find all zeros of the following polynomial functions, noting multiplicities. f(x)=(x3)3(3x1)(x1)2Find all zeros of the following polynomial functions, noting multiplicities. f(x)=2x612x5+18x4Based on the graph, determine the zeros of the function and multiplicities.Use long division to find the quotient. 2x3+3x4x+2Use synthetic division to find the quotient. If the divisor is a factor, write the factored form. x4+3x24x2Use synthetic division to find the quotient. If the divisor is a factor, write the factored form. 2x3+5x27x12x+3Use the Rational Zero Theorem to help you find the zeros of the polynomial functions. f(x)=2x3+5x26x9Use the Rational Zero Theorem to help you find the zeros of the polynomial functions. 14. f(x)=4x4+8x3+21x2+17x+4Use the Rational Zero Theorem to help you find the zeros of the polynomial functions. f(x)=4x4+16x3+13x215x8Use the Rational Zero Theorem to help you find the zeros of the polynomial functions. f(x)=x5+6x4+13x3+14x2+12x+8Given the following information about a polynomial function, find the function. It has a double zero at x=3 and zeroes at x=1 and x=2 It’s y-intercept is (0,12).Given the following information about a polynomial function, find the function. It has a zero of multiplicity 3 at x=12 and another zero at x=3 . It contains the point (1,8).Use Descartes’ Rule of Signs to determine the possible number of positive and negative solutions. 8x321x2+6=0For the following rational functions, find the intercepts and horizontal and vertical asymptotes, and sketch a graph. f(x)=x+4x22x3For the following rational functions, find the intercepts and horizontal and vertical asymptotes, and sketch a graph. f(x)=x2+2x3x24Find the slant asymptote of the rational function. f(x)=x2+3x3x1Find the inverse of the function. f(x)=x2+4Find the inverse of the function. f(x)=3x34Find the inverse of the function. f(x)=2x+33x1Find the unknown value. y varies inversely as the square of x and when x=3,y=2. Find y if x=1.Find the unknown value. 27. y varies jointly with x and the cube root of 2. If when x=2 and z=27,y=12, find y if x=5 and z=8.Solve the following application problem. The distance a body falls varies directly as the square of the time it falls. If an object falls 64 feet in 2 seconds, how long will it take to fall 256 feet?Which of the following equations represent exponential functions? • f(x)=2x23x+1 • g(x)=0.875x • h(x)=1.75x+2 • j(x)=1095.62xLet f(x)=8(1.2)x5. Evaluate f(3) using a calculator. Round to four decimal places.The population of China was about 1.39 billion in the year 2013, with an annual growth rate of about 0.6 . Thissituation is represented by the growth function P(t)=1.39(1.006)t, where tis the number of years since 2013. To thenearest thousandth, what will the population of China be for theyear 2031? How does this compare to the populationprediction we made for India in Example 3?A wolf population is growing exponentially. In 2011, 129 wolves were counted. By 2013, the population had reached 236 wolves. What two points can be used to derive an exponential equation modeling this situation? Write the equation representing the population N of wolves over time t.Given the two points (1,3) and (2,4.5) , find the equation of the exponential function that passes through these twopoints.Find an equation for the exponential function graphed in Figure 6.Use a graphing calculator to find the exponential equation that includes the points (3,75.98) and (6,481.07).An initial investment of 100,000 at 12 interest is compounded weekly (use 52 weeks in a year). What will theinvestment be worth in 30 years?Refer to Example 9. To the nearest dollar, how much would Lily need to invest if the account is compounded quarterly?Use a calculator to find e0.5. Round to five decimal places.A person invests 100,000 at a nominal 12 interest per year compounded continuously. What will be the value of theinvestment in 30 years?Using the data in Example 12, how much radon-222 will remain after one year?Explain why the values of an increasing exponentialfunction will eventually overtake the valuesof anincreasing linear function.Given a formula for an exponential function, is itpossible to determine whether the function grows ordecays exponentiallyjust by looking at the formula?Explain.The Oxford Dictionary defines the word nominal asa value that is “stated or expressed but notnecessarily corresponding exactly to the real value.[18]Develop a reasonable argument for why the termnominal rate is used to describe the annual percentagerate of an investment account that compoundsinterest.For the following exercises, identify whether the statement represents an exponential function. Explain. The average annual population increase of a pack of wolves is 25.For the following exercises, identify whether the statement represents an exponential function. Explain. A population of bacteria decreases by a factor of 18 every 24 hours.For the following exercises, identify whether the statement represents an exponential function. Explain. 6. The value of a coin collection has increased by 3.25 annually over the last 20 years.For the following exercises, identify whether the statement represents an exponential function. Explain. For each training session, a personal trainer charges his clients 5 less than the previous training session.For the following exercises, identify whether the statement represents an exponential function. Explain. 8. The height of a projectile at time tis represented bythe function h(t)=4.9t2+18t+40.For the following exercises, consider this scenario: For each year t, the population of a forest oftrees is representedby the function A(t)=115(1.025)t. In a neighboring forest, the population of the same type of tree is representedby the function B(t)=82(1.029)t. (Round answers to the nearest whole number.) 9. Which forest’s population is growing at a faster rate?For the following exercises, consider this scenario: For each year t , the population of a forest of trees is represented by the function A(t)=115(1.025)t. In a neighboring forest, the population of the same type of tree is represented by the function B(t)=82(1.029)t. (Round answers to the nearest whole number.) Which forest had a greater number of trees initially? By how many?For the following exercises, consider this scenario: For each year t , the population of a forest of trees is represented by the function A(t)=115(1.025)t. In a neighboring forest, the population of the same type of tree is represented by the function B(t)=82(1.029)t. (Round answers to the nearest whole number.) Assuming the population growth models continue to represent the growth of the forests, which forest will have a greater number of trees after 20 years? By how many?For the following exercises, consider this scenario: For each year t , the population of a forest of trees is represented by the function A(t)=115(1.025)t. In a neighboring forest, the population of the same type of tree is represented by the function B(t)=82(1.029)t. (Round answers to the nearest whole number.) Assuming the population growth models continue to represent the growth of the forests, which forest will have a greater number of trees after 100 years? By how many?For the following exercises, consider this scenario: For each year t , the population of a forest of trees is represented by the function A(t)=115(1.025)t. In a neighboring forest, the population of the same type of tree is represented by the function B(t)=82(1.029)t. (Round answers to the nearest whole number.) Discuss the above results from the previous four exercises. Assuming the population growth models continue to represent the growth of the forests, which forest will have the greater number of trees in the long run? Why? What are some factors that might in?uence the long-term validity of the exponential growth model?For the following exercises, determine whether the equation represents exponential growth, exponential decay, or neither. Explain. y=300(1t)5For the following exercises, determine whether the equation represents exponential growth, exponential decay, or neither. Explain. y=220(1.06)xFor the following exercises, determine whether the equation represents exponential growth, exponential decay, or neither. Explain. y=16.5(1.025)1xFor the following exercises, determine whether the equation represents exponential growth, exponential decay, or neither. Explain. y=11,701(0.97)tFor the following exercises, find the formula for an exponential function that passes through the two points given. (0,6) and (3,750)For the following exercises, find the formula for an exponential function that passes through the two points given. (0,2000) and (2,20)For the following exercises, find the formula for an exponential function that passes through the two points given. (1,32) and (3,24)For the following exercises, find the formula for an exponential function that passes through the two points given. (2,6) and (3,1)For the following exercises, find the formula for an exponential function that passes through the two points given. (3,1) and (5,4)For the following exercises, determine whether the table could represent a function that is linear, exponential, or neither. If it appears to be exponential, find a function that passes through the points. 23.For the following exercises, determine whether the table could represent a function that is linear, exponential, or neither. If it appears to be exponential, find a function that passes through the points. 24.For the following exercises, determine whether the table could represent a function that is linear, exponential, or neither. If it appears to be exponential, find a function that passes through the points.For the following exercises, determine whether the table could represent a function that is linear, exponential, or neither. If it appears to be exponential, find a function that passes through the points.For the following exercises, determine whether the table could represent a function that is linear, exponential, orneither. If it appears to be exponential, find a function that passes through the points.For the following exercises, use the compound interest formula, A(t)=P(1+rn)nt. After a certain number of years, the value of an investment account is represented by the equation 10,250(1+ 0.04 12)120. What is the value of the account?For the following exercises, use the compound interest formula, A(t)=P(1+rn)nt. What was the initial deposit made to the account in the previous exercise?For the following exercises, use the compound interest formula, A(t)=P(1+rn)nt. How many years had the account from the previous exercise been accumulating interest?For the following exercises, use the compound interest formula, A(t)=P(1+rn)nt. An account is opened with an initial deposit of 6,500 and earns 3.6 interest compounded semi-annually. What Will the account be worth in ? 20 years?For the following exercises, use the compound interest formula, A(t)=P(1+rn)nt. How much more would the account in the previous exercise have been worth if the interest were compounding weekly?For the following exercises, use the compound interest formula, A(t)=P(1+rn)nt. Solve the compound interest formula for the principal, P .For the following exercises, use the compound interest formula, A(t)=P(1+rn)nt. 34. Use the formula found in Exercise #31 to calculatethe initial deposit of an account that is worth 14,472.74 after earning 5.5 interest compoundedmonthly for 5 years. (Round to the nearest dollar.)For the following exercises, use the compound interest formula, A(t)=P(1+rn)nt. 35. How much more would the account in Exercises#31 and #34 be worth if it were earning interest for 5 more years?For the following exercises, use the compound interest formula, A(t)=P(1+rn)nt. Use properties of rational exponents to solve the compound interest formula for the interest rate, r .For the following exercises, use the compound interest formula, A(t)=P(1+rn)nt. Use the formula found in the previous exercise to calculate the interest rate for an account that was compounded semi-annually, had an initial deposit of 9,000 and was worth 13,373.53 after 10 years.For the following exercises, use the compound interest formula, A(t)=P(1+rn)nt. Use the formula found in the previous exercise to calculate the interest rate for an account that was compounded monthly, had an initial deposit of 5,500, and was worth 38,455 after 30 years.For the following exercises, determine whether the equation represents continuous growth, continuous decay, or neither. Explain. y=3742(e)0.75tFor the following exercises, determine whether the equation represents continuous growth, continuous decay, or neither. Explain. y=150(e)3.25tFor the following exercises, determine whether the equation represents continuous growth, continuous decay, or neither. Explain. y=2.25(e)2tFor the following exercises, determine whether the equation represents continuous growth, continuous decay, or neither. Explain. Suppose an investment account is opened with an initial deposit of 12,000 earning 7.2 interest compounded continuously. How much will the account be worth after 30 years?For the following exercises, determine whether the equation represents continuous growth, continuous decay, or neither. Explain. How much less would the account from Exercise 42 be worth after 30 years if it were compounded monthly instead?For the following exercises, evaluate each function. Round answers to four decimal places, if necessary. f(x)=2(5)x, for f(3)For the following exercises, evaluate each function. Round answers to four decimal places, if necessary. f(x)=42x+3, for f(1)For the following exercises, evaluate each function. Round answers to four decimal places, if necessary. f(x)=ex, for f(3)For the following exercises, evaluate each function. Round answers to four decimal places, if necessary. f(x)=2ex1, for f(1)For the following exercises, evaluate each function. Round answers to four decimal places, if necessary. f(x)=2.7(4)x+1+1.5, for f(2)For the following exercises, evaluate each function. Round answers to four decimal places, if necessary. f(x)=1.2e2x0.3, for f(3)For the following exercises, evaluate each function. Round answers to four decimal places, if necessary. f(x)=32(3)x+32, for f(2)For the following exercises, use a graphing calculator to find the equation of an exponential function given the points on the curve (0,3) and (3,375)For the following exercises, use a graphing calculator to find the equation of an exponential function given the points on the curve (3,222.62) and (10,77.456)For the following exercises, use a graphing calculator to find the equation of an exponential function given the points on the curve (20,29.495) and (150,730.89)For the following exercises, use a graphing calculator to find the equation of an exponential function given the points on the curve (5,2.909) and (13,0.005)For the following exercises, use a graphing calculator to find the equation of an exponential function given the points on the curve (11,310.035) and (25,356.3652)The annual percentage yield (APY) of an investmentaccount is a representation ofthe actual interest rateearned on a compounding account. It is based on acompounding period of one year. Show that the APYof an account that compounds monthly can be foundwith the formula APY=(1+r12)121.Repeat the previous exercise to find the formula forthe APY of an account that compounds daily. Usethe results from this and the previous exercise todevelop a function I(n)for the APY of any accountthat compounds n times per year.Recall that an exponential function is any equationwritten in the form f(x)=abx such that a and b arepositive numbers and b1. Any positive numberb; can be written as b=en for some value of n. Usethis fact to rewrite the formula for an exponentialfunction that uses the number e as a base.In an exponential decay function, the base of theexponent is a value between 0 and 1. Thus, for somenumber b1, the exponential decay function canbe written as f(x)=a(1b)x. Use this formula, alongwith the fact that b=en, to show that an exponential decay function takes theform f(x)=a(e)nx for somepositive number n.The formula for the amount A in an investmentaccount with a nominal interest rate r at any timet is given by A(t)=a(e)rt, where a is the amount ofprincipal initially deposited into an account thatcompounds continuously. Prove that the percentageof interest earned to principal at any time t can becalculated with the formula I(t)=ert1.The fox population in a certain region has an annualgrowth rate of 9 per year. In the year 2012, therewere 23,900 fox counted in the area. What is the foxpopulation predicted to be in the year 2020 ?A scientist begins with 100 milligrams of aradioactive substance that decays exponentially. After 35 hours, 50 mg of the substance remains. How manymilligrams will remain after 54 hours?In the year 1985, a house was valued at 110,000. Bythe year 2005, the value hadappreciated to 145,000. What was the annual growth rate between 1985 and 2005 ? Assume that the value continued to grow by thesame percentage. What was the value of the house inthe year 2010 ?A car was valued at 38,000 in the year 2007. By 2013, the value had depreciated to 11,000 If the car’s valuecontinues to drop by the same percentage, what willit be worth by 2017 ?Jamal wants to save 54,000 for a down paymenton a home. How much will he need to invest in anaccount with 8.2 APR, compounding daily, in orderto reach his goal in 5 years?Kyoko has 10,000 that she wants to invest. Her bankhas several investment accounts tochoose from, allcompounding daily. Her goal is to have 15,000 bythe time she finishes graduate school in 6 years. Tothe nearest hundredth of a percent, what should herminimum annual interest rate be in order to reach hergoal? (Hint: solve the compound interest formula forthe interest rate.)Alyssa opened a retirement account with 7.25 APRin the year 2000. Her initial deposit was 13,500. How much will the account be worth in 2025 if interest compounds monthly? How much more wouldshe make if interest compounded continuously?An investment account with an annual interest rateof 7 was opened with an initial deposit of 4,000 Compare the values of the account after 9 yearswhen the interest is compounded annually, quarterly,monthly, and continuously.Sketch the graph of f(x)=4x. State the domain, range, and asymptote.Try it # 2 Graph f(x)=2x1+3. State domain, range, and asymptote.Solve 4=7.85(1.15)x2.27 graphically. Round to the nearest thousandth.Sketch the graph of f(x)=12(4)x. State the domain, range, and asymptote.Find and graph the equation for a function, g(x) , that reflects f(x)=1.25x about the y-axis. State its domain, range, andasymptote.Write the equation for function described below. Give the horizontal asymptote, the domain, andthe range. • f(x)=ex is compressed vertically by a factor of 13, reflected across the x-axis and then shifted down 2 units.What role does the horizontal asymptote of anexponential function play in telling us about the endbehavior of the graph?What is the advantage of knowing how to recognizetransformations of the graph of a parent functionalgebraically?The graph of f(x)=3x is reflected about the y-axisand stretched verticallyby a factor of 4. What isthe equation of the new function, g(x) ? State itsy-intercept, domain, and range.The graph of f(x)=(12)x is reflected about they-axis and compressed vertically by a factor of 15. What is the equation of the new function, g(x) ? Stateits y-intercept, domain, and range.The graph of f(x)=10x is reflected about the x-axisand shifted upward 7 units. What is the equation ofthe new function, g(x) ? State its y-intercept, domain,and range.The graph of f(x)=(1.68)x is shifted right 3 units,stretched vertically by a factor of 2, reflected aboutthe x-axis, and then shifted downward 3 units. Whatis the equation of the new function, g(x) ? State itsy-intercept (to the nearest thousandth), domain, andrange.The graph of f(x)=2(14)x20 is shifted downward 4 units, and then shifted left 2 units,stretched verticallyby a factor of 4, and reflected about the x-axis. Whatis the equation of the newfunction, g(x) ? State itsy-intercept, domain, and range.For the following exercises, graph the function and its re?ection about the y -axis on the same axes, and give the y -intercept. f(x)=3(12)xFor the following exercises, graph the function and its re?ection about the y -axis on the same axes, and give the y-intercept. g(x)=2(0.25)xFor the following exercises, graph the function and its re?ection about the y -axis on the same axes, and give the y-intercept. h(x)=6(1.75)xFor the following exercises, graph each set of functions on the same axes. f(x)=3(14)x,g(x)=3(2)x, and h(x)=3(4)xFor the following exercises, graph each set of functions on the same axes. f(x)=14(3)x,g(x)=2(3)x, and h(x)=4(3)xFor the following exercises, match each function with one of the graphs in Figure 12. f(x)=2(0.69)xFor the following exercises, match each function with one of the graphs in Figure 12. 14. f(x)=2(1.28)xFor the following exercises, match each function with one of the graphs in Figure 12.For the following exercises, match each function with one of the graphs in Figure 12. f(x)=4(1.28)xFor the following exercises, match each function with one of the graphs in Figure 12. f(x)=2(1.59)xFor the following exercises, match each function with one of the graphs in Figure 12. f(x)=4(0.69)xFor the following exercises, use the graphs shown in Figure 13. All have the form f(x)=abx. Which graph has the largest value for b ?For the following exercises, use the graphs shown in Figure 13. All have the form f(x)=abx. Which graph has the smallest value for b ?For the following exercises, use the graphs shown in Figure 13. All have the form f(x)=abx. Which graph has the largest value for a ?For the following exercises, use the graphs shown in Figure 13. All have the form f(x)=abx. Which graph has the smallest value for a?For the following exercises, graph the function and its re?ection about the x -axis on the same axes. f(x)=12(4)xFor the following exercises, graph the function and its re?ection about the x -axis on the same axes. f(x)=3(0.75)x1For the following exercises, graph the function and its reflection about the x-axis on the same axes. 25. f(x)=4(2)x+2For the following exercises, graph the transformation of f(x)=2x. Give the horizontal asymptote, the domain, and the range. f(x)=2xFor the following exercises, graph the transformation of f(x)=2x. Give the horizontal asymptote, the domain, and the range. h(x)=2x+3For the following exercises, graph the transformation of f(x)=2x. Give the horizontal asymptote, the domain, and the range. f(x)=2x2For the following exercises, describe the end behavior of the graphs of the functions. f(x)=5(4)x1For the following exercises, describe the end behavior of the graphs of the functions. f(x)=3(12)x2For the following exercises, describe the end behavior of the graphs of the functions. f(x)=3(4)x+2For the following exercises, start with the graph of f(x)=4x. Then write a function that results from the given transformation. Shift f(x)4 units upwardFor the following exercises, start with the graph of f(x)=4x. Then write a function that results from the given transformation. Shift f(x)3 units downwardFor the following exercises, start with the graph of f(x)=4x. Then write a function that results from the given transformation. Shift f(x)2 units leftFor the following exercises, start with the graph of f(x)=4x. Then write a function that results from the given transformation. Shift f(x)5 units rightFor the following exercises, start with the graph of f(x)=4x. Then write a function that results from the given transformation. Re?ect f(x) about the x -axisFor the following exercises, start with the graph of f(x)=4x. Then write a function that results from the given transformation. Re?ect f(x) about the y -axisFor the following exercises, each graph is a transformation of y=2x. Write an equation describing the transformation.For the following exercises, each graph is a transformation of y=2x. Write an equation describing the transformation.For the following exercises, each graph is a transformation of y=2x. Write an equation describing the transformation.For the following exercises, find an exponential equation for the graph.For the following exercises, find an exponential equation for the graph.For the following exercises, evaluate the exponential functions for the indicated value of x . g(x)=13(7)x2 for g(6).For the following exercises, evaluate the exponential functions for the indicated value of x . f(x)=4(2)x12 for f(5).For the following exercises, evaluate the exponential functions for the indicated value of x . h(x)=12(12)x+6 for h(7).For the following exercises, use a graphing calculator to approximate the solutions of the equation. Round to the nearest thousandth. f(x)=abx+d. 50=(12)xFor the following exercises, use a graphing calculator to approximate the solutions of the equation. Round to the nearest thousandth. f(x)=abx+d. 116=14(18)xFor the following exercises, use a graphing calculator to approximate the solutions of the equation. Round to the nearest thousandth. f(x)=abx+d. 12=2(3)x+1For the following exercises, use a graphing calculator to approximate the solutions of the equation. Round to the nearest thousandth. f(x)=abx+d. 5=3(12)x12For the following exercises, use a graphing calculator to approximate the solutions of the equation. Round to the nearest thousandth. f(x)=abx+d. 30=4(2)x+2+2Explore and discuss the graphs of f(x)=(b)x and g(x)=(1b)x. Then make a conjecture about therelationship between the graphs of the functions bx and (1b)x for any real number b0.Prove the conjecture made in the previous exercise.Explore and discuss the graphs of f(x)=4x,g(x)=4x2, and h(x)=(116)4x. Then make aconjecture about the relationship between the graphsof the functions bx and (1bn)bx for any real number nand real number b0.Prove the conjecture made in the previous exercise.Write the following logarithmic equations in exponential form. log10(1,000,000)=6 log5(25)=2Write the following exponential equations in logarithmic form. 32=9 53=125 21=12Solve y=log121(11) without using a calculator.Evaluate y=log2(132) without using a calculator.Evaluate y=log(1,000,000).Evaluate y=log(123) to four decimal places using a calculator.The amount of energy released from one earthquake was 8,500 times greater than the amount of energy released fromanother. The equation 10x=8500 represents this situation, where x is thedifference in magnitudes on the Richter Scale.To the nearest thousandth, what was the difference in magnitudes?Evaluate ln(500).What is a base b logarithm? Discuss the meaning byinterpreting each part of the equivalent equations by=x and logb(x)=y for b0,b1.How is the logarithmic function f(x)=logb(x) related to the exponential function g(x)=bx ? Whatis the result of composing these two functions?How can the logarithmic equation logbx=y besolved for x using the properties of exponents?Discuss the meaning of the common logarithm.What is its relationship to a logarithm with base b,and how does the notation differ?Discuss the meaning ofthe natural logarithm. Whatis its relationship to a logarithm with base b, andhow does the notation differ?For the following exercises, rewrite each equation in exponential form. log4(q)=mFor the following exercises, rewrite each equation in exponential form. loga(b)=cFor the following exercises, rewrite each equation in exponential form. log16(y)=xFor the following exercises, rewrite each equation in exponential form. logx(64)=yFor the following exercises, rewrite each equation in exponential form. logy(x)=11For the following exercises, rewrite each equation in exponential form. log15(a)=bFor the following exercises, rewrite each equation in exponential form. logy(137)=xFor the following exercises, rewrite each equation in exponential form. log13(142)=aFor the following exercises, rewrite each equation in exponential form. log(v)=tFor the following exercises, rewrite each equation in exponential form. ln(w)=nFor the following exercises, rewrite each equation in logarithmic form. 4x=yFor the following exercises, rewrite each equation in logarithmic form. cd=kFor the following exercises, rewrite each equation in logarithmic form. m7=nFor the following exercises, rewrite each equation in logarithmic form. 19x=yFor the following exercises, rewrite each equation in logarithmic form. 20. x1013=yFor the following exercises, rewrite each equation in logarithmic form. n4=103For the following exercises, rewrite each equation in logarithmic form. (75)m=nFor the following exercises, rewrite each equation in logarithmic form. yx=39100For the following exercises, rewrite each equation in logarithmic form. 10a=bFor the following exercises, rewrite each equation in logarithmic form. ek=hFor the following exercises, solve for x by converting the logarithmic equation to exponential form. log3(x)=2For the following exercises, solve for x by converting the logarithmic equation to exponential form. log2(x)=3For the following exercises, solve for x by converting the logarithmic equation to exponential form. log5(x)=2For the following exercises, solve for x by converting the logarithmic equation to exponential form. log3(x)=3For the following exercises, solve for x by converting the logarithmic equation to exponential form. log2(x)=6For the following exercises, solve for x by converting the logarithmic equation to exponential form. log9(x)=12For the following exercises, solve for x by converting the logarithmic equation to exponential form. log18(x)=2For the following exercises, solve for x by converting the logarithmic equation to exponential form. log6(x)=3For the following exercises, solve for x by converting the logarithmic equation to exponential form. log(x)=3For the following exercises, solve for x by converting the logarithmic equation to exponential form. ln(x)=2For the following exercises, use the definition of common and natural logarithms to simplify. log(1008)For the following exercises, use the definition of common and natural logarithms to simplify. 10log(32)For the following exercises, use the definition of common and natural logarithms to simplify. 2log(0.0001)For the following exercises, use the definition of common and natural logarithms to simplify. eln(1.06)For the following exercises, use the definition of common and natural logarithms to simplify. ln(e5.03)For the following exercises, use the definition of common and natural logarithms to simplify. 41. eln(10.125)+4For the following exercises, evaluate the base b logarithmic expression without using a calculator. log3(127)For the following exercises, evaluate the base b logarithmic expression without using a calculator. log6(6)For the following exercises, evaluate the base b logarithmic expression without using a calculator. log2(18)+4For the following exercises, evaluate the base b logarithmic expression without using a calculator. 6log8(4)For the following exercises, evaluate the common logarithmic expression without using a calculator. log(10,000)For the following exercises, evaluate the common logarithmic expression without using a calculator. log(0.001)For the following exercises, evaluate the common logarithmic expression without using a calculator. 48. log(1)+7For the following exercises, evaluate the common logarithmic expression without using a calculator. 2log(1003)For the following exercises, evaluate the natural logarithmic expression without using a calculator. ln(e13)For the following exercises, evaluate the natural logarithmic expression without using a calculator. ln(1)For the following exercises, evaluate the natural logarithmic expression without using a calculator. ln(e0.225)3For the following exercises, evaluate the natural logarithmic expression without using a calculator. 25ln(e25)For the following exercises, evaluate each expression using a calculator. Round to the nearest thousandth. log(0.04)For the following exercises, evaluate each expression using a calculator. Round to the nearest thousandth. ln(15)For the following exercises, evaluate each expression using a calculator. Round to the nearest thousandth. ln(45)For the following exercises, evaluate each expression using a calculator. Round to the nearest thousandth. log(2)For the following exercises, evaluate each expression using a calculator. Round to the nearest thousandth. ln(2)Is x=0 in the domain of the function f(x)=log(x) ?If so, what is the value of the function when x=0 fiverify the result.Is f(x)=0 in the range ofthe function f(x)=log(x) ?If so, for what value ofx? Verify the result.Is there a number x such that ln x=2 ? If so, whatis that number? Verify the result.Is the following true: log3(27)log4(1 64)=1 ? Verify the result.Is the following true: ln(e1.725)ln(1)=1.725 ? Verify theresult.The exposure index E1 for a 35 millimeter camera is ameasurement of the amount of light that hits the ?lm.It is determined by the equation EI=log2(f2t),wheref is the f-stop" setting on the camera, and t is theexposure time in seconds. Suppose the f-stop settingis 8 and the desired exposure time is 2 seconds. Whatwill the resulting exposure index he?Refer to the previous exercise. Suppose the lightmeter on a camera indicates an EI of 2 , and thedesired exposure time is 16 seconds. What should thef-stop setting be?The intensitylevels I of two earthquakes measured ona seismograph can be compared by the formula logI1I2=M1M2 where M is the magnitude given by the Richter Scale. In August 2009, an earthquake of magnitude 6.1 hitHonshu, Japan. In March 2011, that same regionexperienced yet another, more devastating earthquake, this time with a magnitude of 9.0.[23] Haw many timesgreater was the intensity of the 2011 earthquake?Round to the nearest whole number.What is the domain of f(x)=log5(x2)+1 ?What is the domain of f(x)=log(x5)+2 ?Graph f(x)=log15(x). State the domain, range, and asymptote.Sketch a graph of f(x)=log3(x+4) alongside its parent function. Include the key points and asymptotes on the graph.State the domain, range, and asymptote.Sketch a graph of f(x)=log2(x)+2 alongside its parent function. Include the key points and asymptote on the graph.State the domain, range, and asymptote.Sketch a graph of f(x)=12log4(x) alongside its parent function. Include the key points and asymptote on the graph. Statethe domain, range, and asymptote.Sketch a graph of the function f(x)=3log(x2)+1. State the domain, range, and asymptote.Graph f(x)=log(x). State the domain, range, and asymptote.Solve 5log(x+2)=4log(x) graphically. Round to the nearest thousandth.What is the vertical asymptote of f(x)=3+ln(x1) ?Give the equation ofthe natural logarithm graphed in Figure 16.The inverse of every logarithmic function is anexponential function and vice-versa. What doesthis tell us about the relationship between thecoordinates of the points on the graphs ofeach?What type (s) of translation(s), if any, affect the range of a logarithmic function?What type (s) of translation (s), if any, affect thedomain ofa logarithmic function?Consider the general logarithmic function f(x)=logb(x). Why can’t x be zero?Does the graph of a general logarithmic functionhave a horizontal asymptote? Explain.For the following exercises, state the domain and range of the function. f(x)=log3(x+4)For the following exercises, state the domain and range of the function. h(x)=ln(12x)For the following exercises, state the domain and range of the function. g(x)=log5+(2x+9)2For the following exercises, state the domain and range of the function. h(x)=ln(4x+17)5For the following exercises, state the domain and range of the function. f(x)=log2(123x)3For the following exercises, state the domain and the vertical asymptote of the function. 11. f(x)=logb(x5)For the following exercises, state the domain and the vertical asymptote of the function. 12. g(x)=ln(3x)For the following exercises, state the domain and the vertical asymptote of the function. 13. f(x)=log(3x+1)For the following exercises, state the domain and the vertical asymptote of the function. f(x)=3log(x)+2For the following exercises, state the domain and the vertical asymptote of the function. g(x)=ln(3x+9)7For the following exercises, state the domain, vertical asymptote, and end behavior of the function. f(x)=ln(2x)For the following exercises, state the domain, vertical asymptote, and end behavior of the function. f(x)=log(x37)For the following exercises, state the domain, vertical asymptote, and end behavior of the function. h(x)=log(3x4)+3For the following exercises, state the domain, vertical asymptote, and end behavior of the function. g(x)=ln(2x+6)5For the following exercises, state the domain, vertical asymptote, and end behavior of the function. f(x)=log3(155x)+6For the following exercises, state the domain, range, and x - and y -intercepts, if they exist. If they do not exist, write DNE. h(x)=log4(x1)+1For the following exercises, state the domain, range, and x - and y -intercepts, if they exist. If they do not exist, write DNE. f(x)=log(5x+10)+3For the following exercises, state the domain, range, and x - and y -intercepts, if they exist. If they do not exist, write DNE. g(x)=ln(x)2For the following exercises, state the domain, range, and x -and y -intercepts, if they exist. If they do not exist, write DNE. f(x)=log2(x+2)5For the following exercises, state the domain, range, and x -and y -intercepts, if they exist. If they do not exist, write DNE. h(x)=3ln(x)9For the following exercises, match each function in Figure 17 with the letter corresponding to its graph.