Precalculus Worksheet 3.1 and 3.2 1. When we first start writing eponential equations, we write them as y= a. However, that isn t the only way an eponential equation can e written. Graph the following three equations on your calculator, then eplain why you see what you see. Show algeraically why this is true. a) y 0.07t = 2e ) 2( 1.07251) t y = c) y = ( ) 9.9 22 t 2. A mechanical engineer earned a yearly salary of $50,000 in 1990 and has averaged a 6.2% raise annually for the last 10 years and epects that this increase will continue. a) Write an equation that models this situation. Let S = yearly salary and n = numer of yrs since 1990. ) According to your equation, what was the engineer s salary in 1980? c) How long will it take for the engineer s salary to reach $100,000? d) Write a douling time equation for this situation. 3. A headache medicine is eliminated from the loodstream at a rate of 12% per hour. Suppose you take a 20mg talet at 4:00 pm. a) Write an equation that models this situation. Be sure to define your variales. ) How many milligrams of medicine are left in the loodstream at 5:30 pm? c) How many hours will it take for half the medicine to e eliminated? d) Write an equation that models the half-life of the medication. 4. The amount C in grams of caron-14 present in a sustance after t years is given y C e - 0.0001216t = 20. a) Find the half-life of caron-14. ) Rewrite the equation for C in the forms y 1 = a and y= a( 2 ) h
5. The half-life of a certain radioactive sustance is 65 days. There are 3.5 grams initially. a) Write an equation that models this situation. Be sure to define your variales. ) How long until there is only 0.5 grams remaining? 6. Often environmentalists will capture an endangered species and transport the species to a controlled environment where the species can produce offspring and regenerate its population (ever watch Rio?). Suppose that eight American ald eagles are captured, transported to Montana, and set free. Based on eperience, the environmentalists epect the population to grow according to the model 500 Pt () = 0.162t 1 + 61.5e - where t is measured in years. a) According to the model, what is the initial population of ald eagles? ) According to the model, what will the population of ald eagles e in 3 years? c) The carrying capacity of the environment is the maimum sustainale numer of ald eagles for the area. What is the carrying capacity of the environment? d) How long does it take for the population to reach one-half of the carrying capacity? 788 7. The spread of the zomie infection in a small town is modeled y the logistic equation Z() t = 0.8t 1 16e, + - where Z is the numer of infected zomies and t is the time in days. a) How many people live in the town? ) How many zomies were there initially? c) How long until half the town is infected?
8. Write the equation of the logistic function whose graph is shown elow. (0, 5) (2, 10) 9. Write the equation of the logistic function that satisfies the following conditions: initial value = 12, limit to growth = 60, passes through the point (1, 24).
Precalculus Worksheet 3.3and 3.4 For questions 1 and 2, rewrite each eponential epression as a logarithm. 1. 5 2 32 2. 3 (0.5) 0.125 For questions 3 and 4, rewrite each logarithmic epression as an eponent. 3. ln148.41 5 4. log 32 5 1/ 2 For questions 5-11, evaluate each logarithmic epression without using a calculator. Show work for 5-8. 5. log36 6 6. log 0.01 7. log 1 64 32 8. log729 1 81 9. log 4 1 10. ln 1.73 e 11. log ( 3 a 4 ) a In prolems 5-8, we evaluated logarithms y putting them into eponential form and getting the same ase. Once the ases were the same, we could set the eponents equal and continue to solve. Use this same strategy on the logarithm prolems 12-13. You may need to condense your logarithms first. 12. 3 3 3 log 5 log log 2 13. 3log 2 log log (24 2 ) 5 5 5 14. If f ( ) = 23-1, find f ( ).
15. The Beer-Lamert law of asorption applied to Lake Erie states that the light intensity I (in lumens) at a depth æ I ö of feet, satisfies the equation logç =-0.00235 çè12. Find the intensity of the light at a depth of 30 feet ø algeraically. 16. Seismologists use the Richter scale to epress the energy, or magnitude, of an earthquake. The Richter æ E ö magnitude of an earthquake, M, is related to the energy released in ergs E shown y the formula 3 M = log 2 ç 10 11.8 çè ø. a) In 1964, an earthquake centered at Prince William Sound, Alaska, registered a magnitude of 9.2 on the Richter scale. This was the largest US earthquake recorded. Find the energy released y the earthquake. ) An encyclopedia gives the equation relating the energy E released and Richter magnitude M of an earthquake as log E 11.8 1.5M. Show this equation is equivalent to the equation aove. Use the properties of logarithms to epand the epression. Simplify, if possile. 1/3 5 6 6 17. log3 81 3 y 18. ln 5 eh In 19-20, condense the epression to write the logarithm of a single epression. Simplify, if possile. 19. ln18 2ln 6 ln 4 20. log 5 3log log y
For 21 22, let log 3 and y log 4. Epand the logarithm. Then, write each epression in terms of and y. 21. 9 log 4 22. log 0.75 23. Evaluate each logarithm with your calculator. Eplain why your answer is reasonale. No, the reason is not ecause the calculator said so! a) log 29 ) log 3 0.4 5.2 24. Prove the Quotient Property of Logarithms. (HINT: In the video, we started y letting log M and y log N. ) log M log M log N N
Pre Calculus Worksheet 3.5 Two algeraic ways to solve an eponential equation 1. Take the log of oth sides of the eponential equation: 1.05 2. Use the properties of logarithms to solve for. 2. Put the equation 1.05 2 into logarithmic form. Use the change of ase formula to solve for. What do you notice aout your answers to questions 1 and 2? 3. Sylvia invests $350 into a savings account earning 5% annual interest. How long does it take Sylvia s money to reach doule the original amount? What do you notice aout this prolem? For #4-12, solve each equation algeraically showing all work. Round final answers to the thousandths. Rememer to check your answers! 4 4. 8 20(0.5) 5. 1 9 5 72 6. 4 1 2 6e 43 7. log (1 2 ) 3 5 2ln 6 8. 9. 7 4 9log 5 3 1 7 log 5 log 1 2 11. log ( a 2) log ( a3) 2 12. log ( ) 2 log ( 10) 3 3 4 4 10.
13. If f( ) 3log ( 1), find 4 f 1 ( ). 14. The radioactive isotope Germanium-71 decays at a rate of 6% per day. A scientist has an initial amount of 15 grams. How long will it take for half the same to remain? Solve algeraically. 15. If a single pane of glass oliterates 10% of the light passing through it, then the percent P of light that passes 0.1n through n successive panes is given approimately y the equation P= 100e -. (Rememer this?) a) Find the inverse of this equation which gives n panes as a function of the percent P of light passing through the window. ) How many panes of glass are needed to lock out 75% of the light? Solve algeraically. 16. Rocket Velocity: Disregarding the force of gravity, the maimum velocity, v, of a rocket is given y v=t ln M where t is the velocity of the ehaust and M is the ratio of the mass of the rocket with fuel to its mass without fuel. A solid propellant rocket has an ehaust velocity of 2.5 kilometers per second. Its maimum velocity is 7.5 kilometers per second. Find its mass ratio, M. If you need additional practice solving logarithm and eponential equations, go ack to the prolems on ws 3.1 & 3.2 where you found the intersection with the calculator and solve them algeraically.