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All Textbook Solutions for Algebra and Trigonometry (MindTap Course List)
38E39Elogx is the exponent to which the base 10 must be raised to get __________. So we can complete the following table for logx. x 103 102 101 100 101 102 103 101/2 logx2E3E4E5E6E7E78 Logarithmic and Exponential forms Complete the table by finding the appropriate logarithmic or exponential form of the equation as in Example 1. Logarithmic form Exponential form 43=64 log42=12 43/2=8 log4(116)=2 log4(12)=12 45/2=1329E10E11E12E13E14E15E16E17E18E19E20E21E22E23E24E25E26E27E28E29E30E31E32E33E34E35E36E37E38E39E40E41E42E43E44E45E4548 Evaluating Logarithms Use a calculator to evaluate the expression, correct to four decimal places. a log50 b log2 c log(32)47E48E49E50E51E52E53E54E55E56E57E58E59E60E61E62E63E64E65E66E67E68E69E70E71E72E73E74E75E76E77E78E79E80E81E82E83E84E85E86E87E88E89E90E91E92E93E94E95E96E97E98E99E100EDifficulty of a Task The difficulty in "acquiring a target" such as using your mouse to click on an icon on your computer screen depends on the distance to the target and the size of the target. According to Fittss Law, the index of difficulty ID is given by ID=log(2A/W)log2 where W is the width of the target and A is the distance to the center of the target. Compare the difficulty of clicking on an icon that is 5 mm wide to clicking on one that is 10 mm wide. In each case, assume that the mouse is 100 mm from the icon.102EDISCUSS: The Googolplex A googol is 10100, and a googolplex is 10googol Find log(log(googol))andlog(log(log(googolplex))104EDICUSS DISCOVER: The Number of Digits in an integer Compare log1000 to the number of digits in 1000. Do the same for 10,000. How many digits does any number between 1000and10, 000 have? Between what two values must the common logarithm of such a number lie? Use your observations to explain why the number of digits in any positive integer xislogx+1. The symbol n is the greatest integer function defined in Section 2.2. How many digits does the number 2100 have?The logarithm of a product of two numbers is the same as the __________ of the logarithms of these numbers. So log5(25.125)= __________ __________.2E3E4E5Ea Most calculators can find logarithms with base __________ and base ________. To find logarithms with different bases, we use the _____________ Formula. To find log7 12, we write log712=loglog b Do we get the same answer if we perform the calculation in part a using ln in place of log?7E8E9E10E11E12E13E14E15E16E17E18E19E20E21E22E23E24E25E26E27E28E29E30E31E32E33E34E35E36E37E38E39E40E41E2348 Expanding Logarithmic Expressions Use the Laws of Logarithms to expand the expression. logx2+4343E44E45E46E47E48E49E50E51E52E53E54E55E56E57E58E59E60E61E62E63E64E65E66E67E68E69E70E71E72EWealth Distribution Vilfredo Pareto (18481923) observed that most of the wealth of a country is owned by a few members of the population. Paretos Principle is logP=logcklogW where W is the wealth level how much money a person has and P is the number of people in the population having that much money. aSolve that equation for P. bAssume that k=2.1 and c=8000, and that W is measured in millions of dollars. Use part a to find the number of people who have 2 million or more. How many people have 10 million or more?74EMagnitude of Stars The magnitude M of a star is a measure of how bright a star appears to the human eye. It is defined by M=2.5log(BB0) where B is the actual brightness of the star and B0 is a constant. aExpand the right-hand side of the equation. bUse part a to show that the bright a star, the less its magnitude. cBetelgeuse is about 100 times brighter than Albiero. Use part a to show that Betelgeuse is 5 magnitudes less bright than Albiero.76E77E78ELets solve the exponential equation 2ex=50. a First, we isolate ex to get the equivalent equation _____________. b Next, we take In of each side to get the equivalent equation _________________. c Now we use a calculator to find x.2E3E4E5E6E7E8E9E10E11E12E13E14E15E16E17E18E19E20E1138 Exponential Equations a Find the exact solution of the exponential equation in terms of logarithms. b Use a calculator to find an approximation to the solution rounded to six decimal places. e14x=222E23E24E25E26E27E28E29E30E31E32E33E34E35E36E37E38E39E40E41E42E43E44E45E46E47E48E49E50E51E52E53E54E55E56E57E58E59E60E61E62E63E64E65E66E67E68E69E70E71E72E73E74E75E76E77E78E79E80E81E82E83E84E85E86E87E88E89E90E91E92E93E94E95E96EFish Population A small lake is stocked with a certain species of fish. The fish population is modeled by the function. P=101+4e0.8t where P is the number of fish in thousands and t is measured in years since the lake was stocked. a Find the fish population after 3 years. b After how many years will the fish population reach 5000 fish?98E99ECooling an Engine Suppose youre driving your car on a cold winter day 20F outside and the engine overheats at about 220F. When you park, the engine begins to cool down. The temperature T of the engine t minutes after you park satisfies the equation ln(T20200)=0.11t a Solve the equation for T. b Use part a to find the temperature of the engine after 20min(t=20).Electric circuits An electric circuit contains a battery that produces a voltage of 60 volts V, a resistor with a resistance of 13 ohms (), and an inductor with an inductance of 5 henrys H, as shown in the figure on the following page. Using calculus, it can be shown that the current I=I(t) in amperes, A t seconds after the switch is closed is I=6013(1e13t/5). a Use this equation to express the time t as a function of the current I. b After how many seconds is the current 2 A?Learning Curve A learning curve is a graph of a function P(t) that measures the performance of someone learning a skill as a function of the training time t. At first, the rate of learning is rapid. Then, as performance increases and approaches a maximal value M, the rate of learning decreases. It has been found that the function P(t)=MCekt where k and C are positive constants and CM is a reasonable model for learning. a Express the learning time t as a function of the performance level P. b For s pole-vaulter in training, the learning curve is given by P(t)=2014e0.024t where P(t) is the height he is able to pole-vault after t months. After how many months of training is he able to vault 12 ft? c Draw a graph of the learning curve in part b.103E104E105E116 Population Growth These exercises use the population growth model. Bacteria Culture A certain culture of the bacterium Streptococcus A initially has 10 bacteria and is observed to double every 1.5 hours. a Find an exponential model n(t)=n02t/a for the number of bacteria in the culture after t hours. b Estimate the number of bacteria after 35 hours. c After how many hours will the bacteria count reach 10, 000?116 Population Growth These exercises use the population growth model. Bacteria Culture A certain culture of the bacterium Rhodo-bacter sphaeroides initially has 25 bacteria and is observed to double every 5 hours. a Find an exponential model n(t)=n02t/a for the number of bacteria in the culture after t hours. b Estimate the number of bacteria after 18 hours. c After how many hours will the bacteria count reach 1 million?3E4E5E6E7E8E9E116 Population Growth These exercises use the population growth model. Bat Population The bat population in a certain Midwestern county was 350, 000 in 2012, and the observed doubling time for the population is 25 years. a Find an exponential model n(t)=n02t/a for the population t years after 2012. b Find an exponential model n(t)=n0ert for the population t years after 2012. c Sketch a graph of the population at time t. d Estimate how long it takes the population to reach 2 million.