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The Natural LogName_________________________Base of a natural log = ___________ The natural log of x is written: _________________*The same properties and rules of logs apply to the natural log!Express each in log form. 1. e-2≈0.1353 2. em=n3. e0 = 1Express each in exponential form.4. ln4.2=x 5. ln14≈-1.3863 6. lne=1Evaluate without a calculator.7. lne8. lne59. ln?(-5)10. eln4Use the properties of logarithms to express as a single log.11. ln48-4ln2 12. 12ln9+ln12-2ln3 13. -13ln8 + 3Expand using the properties of logarithms.14. ln2a7b3 15. ln1ab 16. lnx+yx-y 17. Write the equation of a natural logarithm that has been vertically compressed by 4, reflected across the x-axis, shifted right 6 units, horizontally stretched by 2, and shifted down 3 units. Evaluate without a calculator. Leave answers in exact value form.18. lne5 19. ln1e 20. ln7e4 21. eln6+ln5 22. e13ln8-12ln9 23. 12ln4 + ln8- 5ln2+ln324. ex=5 25. elnx =12 26. x=lne35Solve with a calculator. Round to 3 decimal places if necessary. 24. 220=100e0.06x 25. lnx-1-ln8=226. lnx-4+ln10=527. 1.2e-5x + 2.6 = 328. 101+2e-4x=929. 4e4x-18=50Natural Logarithms HomeworkExpress in logarithmic form:1. e2.5≈12.182. e-2≈0.133. e15≈1.221 4. e≈1.649Express in exponential form:5. ln8≈2.0796. ln0.5≈-0.6937. ln0.1≈-2.3038. ln0.01≈-4.605Simplify WITHOUT a calculator (leave answer as a SINGLE log if necessary):9. lne410. eln211. eln6+ln712. e2ln713. ln48-4ln214. 12ln9+ln12-2ln315. 12(ln45+ln5)-2ln316. eln8-ln6 17. e12ln3 18. ln6+ln30-(ln5+3ln2) 19. 12ln4+ln8-(5ln2+ln3) 20. 3ln4-(ln2+ln8) Applications of Exponentials & LogarithmsName_________________________Formulas for Compound InterestFor n # of times compounding per year: A=P(1+rn)nt A = _____________P = _____________r = ____________ t = __________annual n = ____ semiannual n = _____ monthly n = _____quarterly n = ______daily n = ______ For continuous compounding: A=PertSet up and solve the following problems.1. You invest $500 in a savings account that pays 3.5% annual interest. How much will be in the account after five years?2. You invest $1500 in a savings account that pays 4.5% annual interest compounded semi-annually. How much will be in the account after seven years?3. You invest $2000 in a savings account that pays 3.1% annual interest compounded quarterly. How much will be in the account after four years?4. You invest $2100 in a savings account that pays 4.2% annual interest compounded monthly. How much will be in the account after three years?5. You invest $900 in a savings account that pays 5.5% annual interest compounded daily. How much will be in the account after eight years?6. You invest $1250 in a savings account that pays 6% annual interest compounded continuously. How much will be in the account after ten years?7. You invest $2000 in a savings account that pays 4.7% interest compounded continuously. How long will it take for the account to have $3515.78?8. How long will it take your money to double at 2.4 % compounded continuously?9. A population of insects is growing in such a way that the number in the population t days from now is given by the formula P=4000e0.02t. How large will the population be in one week?Given the original principal, the annual interest rate, the amount of time for each investment, and the type of compounded interest, find the amount at the end of the investment.10. P = $1000, r = 10%, t = 4 years, monthly11. P = $3200, r = 6%, t = 5 years 6 months, quarterly12. P = $750, r = 5.5%, t = 3 years 2 months, continuously13. P = $45,000, r = 7.2%, t = 30 years, daily14. The yield, y, in millions of cubic feet of trees per acre for a forest stand that is t years old is given by y=6.7e-48.1t.a. Find the yield after 15 years.b. Find the yield after 50 years.c. Graph the yield per year on a graphing calculator for 150 years. Does the yield ever decrease? If so, when?d. What is the yield after 150 years?15. One model used by political scientists to predict the number of legislators in the U. S. House of Representatives that will serve continuously is given by the exponential function y=434e-2t25, where y is the number of legislators and t is the number of years since 1965.a. According to this model, how many of the legislators from 1965 were still in office in 1980?b. Predict the number of legislators from 1965 still in office in 1995.c. Graph the number of legislators from 1965 in office the years 1965 to 2060 on a graphing calculator. When does the model predict that all of the members of the 1965 House will be out of office? Does this seem realistic? Hint: Take the year that the predicted number of legislators becomes less than 0.5 to be the year that the last member of the 1965 House leaves office.Round all answers to 3 decimal places.Continuous Growth or decay: y=y0ekt where y0 = initial amount (starting amount)?????????? k = constant of proportionality, t = time ?????????? y = ending amountHalf-Life:? ____________________________________________________________________________16. Radium-226, a common isotope of radium, has a half-life of 1620 years.? Professor Korbel has a 120 g sample of radium-226 in his laboratory.? Use the store key to keep the value of “k” in the calculator as is.a.)? Find the constant of proportionality for radium-226. b.)? How many grams of the 120 gram sample will remain after 100 years?17. The half-life of a radioactive isotope is 9 years. a) Find the constant “k” for a 20 gram sample.b) How many years will it take for the 20 gram sample to be less than 1 gram?18. The half-life of a radioactive substance is 2200 years. How long will it take the substance to have a 10% weight loss?19. Bob invested a sum of money in a certificate of deposit (CD) that pays 1.25% interest compounded continuously. If he made the investment on January 1st, 2002, and the account is worth $10,000 on January 1st, 2014, what was the original amount in the account?20. The town planners of Apex, NC are studying the current sewage system needs of the town. The system they have now can support 70,000 residents. The current population is 40,000 up from 20,000 20 years ago. Assuming that the population will continue to grow at this rate, when will the sewage system need to be updated? Applications HomeworkRound all answers to 4 places except money.1. Radium 226, which is used for cancer treatment and as an ingredient in fluorescent paint, decomposes radioactively. Its half-life is 1800 years. Find the constant k you would use in the decay formula for radium. Use 1 gram as the original amount.2. Mr. Cuthbert invested a sum of money in a certificate of deposit that pays 8% interest compounded continuously. If Mr. Cuthbert made the investment on January 1, 1986 and the account is worth $10,000 on January 1, 2005, what was the original amount in the account?3. Sales of a product under relatively stable market conditions tend to decline at a constant annual rate in the absence of promotional activities. This sales decline can be expressed by the exponential function of the form s=s0e-at, where s is the sales time at time t, t is time in years, s0 is the sales at time t = 0, and “a” is the sales decay constant. Suppose sales of On-Time Watches were 45,000 the first year and 37,000 the second year.a. Find the value of “a” in the equation for this sales decline.b. Find the projected sales for three years from now (t = 3).c. If the trend continues, when would you expect sales to be 15,000 units?4. Mike Wazowski deposited some money in a bank account that earns 5.6% interest compounded continuously.a. How long would it take to double the amount of money in Mr. Wazowski’s account?b. If Mike deposited $2000 initially, how much money will he have after 7 years?5. The atmospheric pressure varies with the altitude above the surface of the Earth. Meteorologists have determined that for altitudes for up to 10 km, the pressure p in millimeters of mercury is given by p=760e-0.125a, where a is the altitude in kilometers.a. What is the atmospheric pressure at an altitude of 3.3 km?b. At what altitude will the atmospheric pressure be 450 m of mercury?6. You invest $1900 into a bank account that earns 2.8% interest. Given the number of times compounded per year, how much will you have after 10 years?Compounded quarterlyb. Compounded monthlyc. Compounded semi-annuallyd. Compounded daily7. A bacterial culture doubles every 2 hours. If the culture started with 24,000 bacteria, how many bacteria will be present in 5 hours?8. The half life of a radioactive sample is 4 hours. If 60 g of the sample was initially present, how much will remain after 7 hours?9. The population of a town triples every 6 years. If 4000 people are present in 2006, how many people will be in the town in 2016?10. The half life of a radioactive sample is 6.2 hours. If 2000 g of the sample is present after 7 hours, how much was initially present?11. A radioactive sample has a half life of 3 days. How long will it take for only 1/8 of the sample to remain?12. $2300 is invested at 6% compounded monthly for 7 years. How much interest is earned?13. An investment is invested at 5% compounded continuously. How many years will it take to triple in value?14. An investment service promises to triple your money in 12 years. Assuming continuous compounding of interest, what rate of interest is needed? ................
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