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mathematics
precalculus 1st

Precalculus 1st Edition Jay Abramson - Solutions

For the following exercises, evaluate each function. Round answers to four decimal places, if necessary.f(x) = −2ex − 1, for f(−1)
Use the one-to-one property of logarithms to find an exact solution for log8 (7) + log8 (−4x) = log8 (5). If there is no solution, write no solution.
For the following exercises, solve each equation for x.log8 (x + 6) − log8 (x) = log8 (58)
For the following exercises, find the value of the number shown on each logarithmic scale. Round all answers to the nearest thousandth.Plot each set of approximate values of intensity of sounds on a logarithmic scale: Whisper: 10−10 W/m2, Vacuum: 10−4 W/m2 , Jet: 102 W/m2
For 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 + 1
For the following exercises, refer to Table 11.To the nearest whole number, what is the predicted carrying capacity of the model? f(x) 1 8.7 2 12.3 3 15.4 4 18.5 5 20.7 Table 11 6 7 22.5 23.3 8 9 10 24 24.6 24.8
For the following exercises, evaluate the common logarithmic expression without using a calculator.log(1) + 7
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)
Use the one-to-one property of logarithms to find an exact solution for ln(5) + ln(5x2 − 5) = ln(56). If there is no solution, write no solution.
For the following exercises, solve each equation for x.ln(3) − ln(3 − 3x) = ln(4)
For the following exercises, write a logarithmic equation corresponding to the graph shown.Use f(x) = log4 (x) as the parent function. نا انا انا 5+ IIIIII IIIIO X
For the following exercises, find the value of the number shown on each logarithmic scale. Round all answers to the nearest thousandth.Recall the formula for calculating the magnitude of an earthquake,One earthquake has magnitude 3.9 on the MMS scale. If a second earthquake has 750 times as much
For the following exercises, use a graphing calculator to approximate the solutions of the equation. Round to the nearest thousandth. f(x) = abx + d. x-1 5= = ³ ( 1²1 ) * - 2 - 2
For the following exercises, evaluate the common logarithmic expression without using a calculator.2 log(100−3)
For the following exercises, evaluate each function. Round answers to four decimal places, if necessary. _f(x) = −(3)-x+ 2 3 2 for f(2)
The formula for measuring sound intensity in decibels D is defined by the equation D = 10log (I/I0), where I is the intensity of the sound in watts per square meter and I0 = 10−12 is the lowest level of sound that the average person can hear. How many decibels are emitted from a large orchestra
For the following exercises, refer to Table 11.Use the intersect feature to find the value of x for which the model reaches half its carrying capacity. x 1 f(x) 8.7 2 3 12.3 15.4 4 18.5 5 20.7 Table 11 6 22.5 7 23.3 780 24 9 10 24.6 24.8
For the following exercises, solve each equation for x.log3 (3x) − log3 (6) = log3 (77)
For 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 + 2
For the following exercises, write a logarithmic equation corresponding to the graph shown.Use f(x) = log5 (x) as the parent function. 54 5+ 3 IIIIIIII X
For the following exercises, use this scenario: The equation N(t) models the number of people in a town who have heard a rumor after t days.How many people started the rumor? 500 1+ 49e-0.7t
The population of a city is modeled by the equation P(t) = 256, 114e0.25t where t is measured in years. If the city continues to grow at this rate, how many years will it take for the population to reach one million?
For the following exercises, solve the equation for x, if there is a solution. Then graph both sides of the equation, and observe the point of intersection (if it exists) to verify the solution.log9 (x) − 5 = −4
For the following exercises, evaluate the natural logarithmic expression without using a calculator. uT ६ (a) I
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, refer to Table 12.Use a graphing calculator to create a scatter diagram of the data. 0 f(x) 12 J 2 28.6 4 5 7 52.8 70.3 99.9 Table 12 17 8 10 11 15 112.5 125.8 127.9 135.1 135.9
For the following exercises, use a graphing calculator to find approximate solutions to each equation.log(x − 1) + 2 = ln(x − 1) + 2
For the following exercises, use this scenario: The equation N(t) models the number of people in a town who have heard a rumor after t days.To the nearest whole number, how many people will have heard the rumor after 3 days? 500 1+ 49e-0.7t
Explore and discuss the graphs of f(x) = (b)x and . Then make a conjecture about the relationship between the graphs of the functions for any real number b > 0. g(x) = ( 1 ) ² b
For the following exercises, evaluate the natural logarithmic expression without using a calculator.ln(1)
Find the inverse function f −1 for the exponential function f(x) = 2 · ex + 1 − 5.
For the following exercises, refer to Table 12.Use the LOGISTIC regression option to find a logistic growth model of the form y = c/1 + ae−bx that best fits the data in the table. x 0 f(x) 12 2 28.6 4 5 52.8 70.3 7 99.9 Table 12 8 10 11 15 17 112.5 125.8 127.9 135.1 135.9
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, solve the equation for x, if there is a solution. Then graph both sides of the equation, and observe the point of intersection (if it exists) to verify the solution.log3 (x) + 3 = 2
For the following exercises, use this scenario: The equation N(t) models the number of people in a town who have heard a rumor after t days.As t increases without bound, what value does N(t) approach? Interpret your answer. 500 1+ 49e-0.7t
For the following exercises, use a graphing calculator to find approximate solutions to each equation.log(2x − 3) + 2 = −log(2x − 3) + 5
For the following exercises, evaluate the natural logarithmic expression without using a calculator.ln(e−0.225) − 3
For the following exercises, refer to Table 12.Graph the logistic equation on the scatter diagram. fx) 0 12 2 28.6 4 52.8 5 70.3 7 99.9 Table 12 8 10 11 15 17 112.5 125.8 127.9 135.1 135.9
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)
Find the inverse function f −1 for the logarithmic function f(x) = 0.25 · log2 (x3 + 1).
For the following exercises, solve the equation for x, if there is a solution. Then graph both sides of the equation, and observe the point of intersection (if it exists) to verify the solution.ln(3x) = 2
For the following exercises, use a graphing calculator to find approximate solutions to each equation.ln(x − 2) = −ln(x + 1)
For the following exercise, choose the correct answer choice.A doctor and injects a patient with 13 milligrams of radioactive dye that decays exponentially. After 12 minutes, there are 4.75 milligrams of dye remaining in the patient’s system. Which is an appropriate model for this situation?a.
For the following exercises, refer to Table 12.To the nearest whole number, what is the predicted carrying capacity of the model? x f(x) 0 12 2 4 5 28.6 52.8 70.3 7 99.9 Table 12 8 10 11 15 17 112.5 125.8 127.9 135.1 135.9
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 this scenario: A doctor prescribes 300 milligrams of a therapeutic drug that decays by about 17% each hour.To the nearest minute, what is the half-life of the drug?
Explore and discuss the graphs of f(x) = 4x, g(x) = 4x − 2, and h(x) = (1/16)4x. Then make a conjecture about the relationship between the graphs of the functions bx and (1/bn)bx for any real number n and real number b > 0.
For the following exercises, evaluate each expression using a calculator. Round to the nearest thousandth.log(0.04)
For the following exercises, solve the equation for x, if there is a solution. Then graph both sides of the equation, and observe the point of intersection (if it exists) to verify the solution.ln(x − 5) = 1
For the following exercises, use a graphing calculator to find approximate solutions to each equation.2ln(5x + 1) =1/2 ln(−5x) + 1
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)
For the following exercises, refer to Table 12.Use the intersect feature to find the value of x for which the model reaches half its carrying capacity. x 0 f(x) 12 2 28.6 4 5 52.8 70.3 7 99.9 Table 12 8 10 11 15 17 112.5 125.8 127.9 135.1 135.9
For the following exercises, use this scenario: A doctor prescribes 300 milligrams of a therapeutic drug that decays by about 17% each hour.Write an exponential model representing the amount of the drug remaining in the patient’s system after t hours. Then use the formula to find the amount of
For the following exercises, use a graphing calculator to find approximate solutions to each equation.1/3 log(1 − x) = log(x + 1) + 1/3
The annual percentage yield (APY) of an investment account is a representation of the actual interest rate earned on a compounding account. It is based on a compounding period of one year. Show that the APY of an account that compounds monthly can be found with the formula APY = (1 + r
For the following exercises, evaluate each expression using a calculator. Round to the nearest thousandthln(15)
Recall that an exponential function is any equation written in the form f(x) = a . bx such that a and b are positive numbers and b ≠ 1. Any positive number b can be written as b = en for some value of n. Use this fact to rewrite the formula for an exponential function that uses the number e as a
Recall that the general form of a logistic equation for a population is given by P(t) = c/1 + ae−bt , such that the initial population at time t = 0 is P(0) = P0 . Show algebraically that c - P(t) P(t) c - Po P 0 -e-bt
For the following exercises, use this scenario: A soup with an internal temperature of 350° Fahrenheit was taken off the stove to cool in a 71°F room. After fifteen minutes, the internal temperature of the soup was 175°F.Use Newton’s Law of Cooling to write a formula that models this
For the following exercises, use this scenario: A soup with an internal temperature of 350° Fahrenheit was taken off the stove to cool in a 71°F room. After fifteen minutes, the internal temperature of the soup was 175°F.How many minutes will it take the soup to cool to 85°F?
For the following exercises, solve the equation for x, if there is a solution. Then graph both sides of the equation, and observe the point of intersection (if it exists) to verify the solution.−7 + log3 (4 − x) = −6
Let b be any positive real number such that b ≠ 1. What must logb 1 be equal to? Verify the result.
For the following exercises, evaluate each expression using a calculator. Round to the nearest thousandth. In(3 5.
Use a graphing utility to find an exponential regression formula f(x) and a logarithmic regression formula g(x) for the points (1.5, 1.5) and (8.5, 8.5). Round all numbers to 6 decimal places. Graph the points and both formulas along with the line y = x on the same axis. Make a conjecture about the
For the following exercises, solve the equation for x, if there is a solution. Then graph both sides of the equation, and observe the point of intersection (if it exists) to verify the solution.ln(4x − 10) − 6 = −5
For the following exercises, evaluate each expression using a calculator. Round to the nearest thousandth. log(√2)
Explore and discuss the graphs of f(x) = log1/2 (x) and g(x) = −log2 (x). Make a conjecture based on the result.
For the following exercises, solve the equation for x, if there is a solution. Then graph both sides of the equation, and observe the point of intersection (if it exists) to verify the solution.log(4 − 2x) = log(−4x)
For the following exercises, evaluate each expression using a calculator. Round to the nearest thousandth.ln(√2)
In an exponential decay function, the base of the exponent is a value between 0 and 1. Thus, for some number b > 1, the exponential decay function can be written asUse this formula, along with the fact that b = en, to show that an exponential decay function takes the form f (x) = a(e)−nx for some
For the following exercises, use this scenario: The equation N(t) = 1200/1 + 199e−0.625t models the number of people in a school who have heard a rumor after t days.How many people started the rumor?
For the following exercises, use this scenario: The equation N(t) = 1200/1 + 199e−0.625t models the number of people in a school who have heard a rumor after t days.To the nearest tenth, how many days will it be before the rumor spreads to half the carrying capacity?
Find the inverse function f −1 (x) for the logistic function f(x) = c/1 + ae−bx. Show all steps.
For the following exercises, solve the equation for x, if there is a solution. Then graph both sides of the equation, and observe the point of intersection (if it exists) to verify the solution.log11(−2x2 − 7x) = log11(x − 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? Verify the result.
Is f(x) = 0 in the range of the function f (x) = log(x)? If so, for what value of x? Verify the result.
The formula for the amount A in an investment account with a nominal interest rate r at any time t is given by A(t) = a(e)rt, where a is the amount of principal initially deposited into an account that compounds continuously. Prove that the percentage of interest earned to principal at any time t
For the following exercises, use this scenario: The equation N(t) = 1200/1 + 199e−0.625t models the number of people in a school who have heard a rumor after t days.What is the carrying capacity?
Use the result from the previous exercise to graph the logistic model P(t) = 20/1 + 4e−0.5t along with its inverse on the same axis. What are the intercepts and asymptotes of each function?
For the following exercises, solve the equation for x, if there is a solution. Then graph both sides of the equation, and observe the point of intersection (if it exists) to verify the solution.ln(2x + 9) = ln(−5x)
Use properties of exponents to find the x-intercepts of the function f (x) = log(x2 + 4x + 4) algebraically. Show the steps for solving, and then verify the result by graphing the function.
Is there a number x such that ln x = 2? If so, what is that number? Verify the result.
A scientist begins with 100 milligrams of a radioactive substance that decays exponentially. After 35 hours, 50 mg of the substance remains. How many milligrams will remain after 54 hours?
The fox population in a certain region has an annual growth rate of 9% per year. In the year 2012, there were 23,900 fox counted in the area. What is the fox population predicted to be in the year 2020?
For the following exercises, enter the data from each table into a graphing calculator and graph the resulting scatter plots. Determine whether the data from the table would likely represent a function that is linear, exponential, or logarithmic.
For the following exercises, solve the equation for x, if there is a solution. Then graph both sides of the equation, and observe the point of intersection (if it exists) to verify the solution.log(x2 + 13) = log(7x + 3)
Is the following true: Verify the result. In(e¹.725) In(1) = 1.725?
In the year 1985, a house was valued at $110,000. By the year 2005, the value had appreciated to $145,000. What was the annual growth rate between 1985 and 2005? Assume that the value continued to grow by the same percentage. What was the value of the house in the year 2010?
Find a formula for an exponential equation that goes through the points (−2, 100) and (0, 4). Then express the formula as an equivalent equation with base e.
The intensity levels I of two earthquakes measured on a seismograph can be compared by the formulawhere M is the magnitude given by the Richter Scale. In August 2009, an earthquake of magnitude 6.1 hit Honshu, Japan. In March 2011, that same region experienced yet another, more devastating
For the following exercises, solve for the indicated value, and graph the situation showing the solution point.An account with an initial deposit of $6,500 earns 7.25% annual interest, compounded continuously. How much will the account be worth after 20 years?
The population of a culture of bacteria is modeled by the logistic equation P(t) = 14, 250/1 + 29e−0.62t, where t is in days. To the nearest tenth, how many days will it take the culture to reach 75% of its carrying capacity?
Jamal wants to save $54,000 for a down payment on a home. How much will he need to invest in an account with 8.2% APR, compounding daily, in order to reach his goal in 5 years?
What is the carrying capacity for a population modeled by the logistic equation P(t) = 250, 000/1 + 499e−0.45t? What is the initial population for the model?
For the following exercises, solve the equation for x, if there is a solution. Then graph both sides of the equation, and observe the point of intersection (if it exists) to verify the solution.ln(x) − ln(x + 3) = ln(6)
A car was valued at $38,000 in the year 2007. By 2013, the value had depreciated to $11,000 If the car’s value continues to drop by the same percentage, what will it be worth by 2017?
The exposure index EI for a 35 millimeter camera is a measurement of the amount of light that hits the film. It is determined by the equation where f is the “f-stop” setting on the camera, and t is the exposure time in seconds. Suppose the f-stop setting is 8 and the desired exposure time
For the following exercises, solve the equation for x, if there is a solution. Then graph both sides of the equation, and observe the point of intersection (if it exists) to verify the solution. 3 log₂ (10) log(x-9) log(44)
For the following exercises, evaluate the base b logarithmic expression without using a calculator. log₂ +4 1 2 8
For the following exercises, use this scenario: A turkey is taken out of the oven with an internal temperature of 165° Fahrenheit and is allowed to cool in a 75° F room. After half an hour, the internal temperature of the turkey is 145° F.To the nearest degree, what will the temperature be after

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