1.In the formula At p 1 r n nt, identify what the letters P, r, n and t stand for. 2. Find the exponential function whose graph is given f(x) = a x 3. State the domain and range of the function (Enter your answers using interval notation.) y = 2 x 6 g(x) = 2 x 4 g(x) = e x 1 5 Find an exponential function for the situation below. 4. A bacteria culture contains 1200 bacteria initially and doubles every hour. 5. An investment of $4000 is deposited into an account in which interest is compounded monthly with r = 5%. (Round your answers to the nearest cent.) 6 $2000 is invested at an interest rate of 4.5% per year, compounded daily. The present value of a sum of money is the amount that must be invested now, at a given rate of interest, to produce the desired sum at a later date. 7. Find the present value of $10,000 if interest is paid at a rate of 9% per year, compounded semiannually, for 5 years. (Round your answer to the nearest cent.) 8. Find the annual percentage yield for an investment that earns 8% per year, compounded monthly. (Round your answer to two decimal places.) 9. Find the annual percentage yield for an investment that earns 8.5% per year, compounded quarterly. (Round your answer to two decimal places.) 10. Your mathematics instructor asks you to sketch a graph of the exponential function f(x) = 2 x for x between 0 and 40, using a scale of 10 units to one inch. What are the dimensions of the sheet of paper you will need to sketch this graph? (Round your answer for the height to two decimal places.) 11. If f(x) = 11 x, show that h 11 1 f x h f x x 11 h h 12. A radioactive substance decays in such a way that the amount of mass remaining after t days is given by the function m(t) = 13e 0.011t where m(t) is measured in kilograms. How much of the mass remains after 40 days? (Round your answer to one decimal place.) 13. If $1000 is invested in an account for which interest is compounded continuously with r=4%, find the amount of the investment at the end of 11 years for the following interest rates.
14. Which of the given interest rates and compounding periods would provide the best investment? 2% per year, compounded continuously 2% per year, compounded semiannually 2% per year, compounded monthly 15. Convert the following to logarithmic form 6 3 = 216. 16. Convert the following to exponential form log 6(36) = 2. 17. Match the logarithmic function with its graph. f(x) = log2(x), f(x) = log2( x), f(x) = log2(x), f(x) = log2( x) 18. Evaluate the expression. (Simplify your answer completely.) 9 log 3, log 27 9 19. Find the function of the form y = loga(x) whose graph is given.
Sketch the graph of function below. Find the domain and range. f(x) = log2(x 4), y = 1 + ln( x) Find the domain of the function. g(x) = log 4(x 2 1), g(x) = ln(x x 2 ) A function f(x) is given f(x) = log6(log8(x)). (a) Find the domain of the function f. (Enter your answer using interval notation.) (b) Find the inverse function of f. (a) Find the inverse of the function x 5 f x 3 5 (b) What is the domain of the inverse function? (Enter your answer using interval notation.) Which is larger, log4(17) or log5(24)? (Justify your answer.) Expand log x y 5 z 4 2 3 using single logarithm. Combine to single logarithm ½ log x + log y 3 log z, 2 log(x) (1/5) log(x 2 + 1) + 4 log(x 1) x Use the Laws of Logarithms to expand the expression log x y z Use the Change of Base Formula and a calculator to evaluate the logarithm, correct to six decimal places. log8(97) Solve the logarithmic/exponential equations. log(3) + log(x 2) = log(x), e 3x = 10, 4 1 x = 9, 5 x = 4 x + 1 5 3. Solve the equation. (Round your answers to four decimal places.) e 2x 5e x + 4 = 0, 2 x 10(2 x ) + 3 = 0, log 2(x + 8) log 2(x 8) = 3 Find the inverse function of f. f(x) = 3 x + 5 f(x) = log 4(x 1) f(x) = 5 6x A woman invests $6300 in an account that pays 6% interest per year. How long will it take for the amount to be $11,000? (Round your answer to two decimal.) If the interest compounded monthly? Continuously? A sum of $1000 was invested for 4 years, and the interest was compounded semiannually. If this sum amounted to $1463.44 in the given time, what was the interest rate? (Round your answer to two decimal places.)
A 15-g sample of radioactive iodine decays in such a way that the mass remaining after t days is given by m(t) = 15e 0.081t. After how many days is there only 5 g remaining? (Round your answer to the nearest whole number.) A certain culture of the bacterium Rhodobacter sphaeroides initially has 45 bacteria and is observed to double every 6 hours. Find an exponential model. A certain culture of the bacterium Streptococcus A initially has 6 bacteria and is observed to double every 1.5 hours.(a) Find an exponential model (b) After how many hours will the bacteria count reach 10,000? (Round your answer to one decimal place.) The fox population in a certain region has a relative growth rate of 6% per year. It is estimated that the population in 2013 was 15,000. (a) Find a function that models the population t years after 2013 as n(t) = n0e rt (b) After how many years will the fox population reach 23,000? (Round your answer to one decimal place.) The graph shows the deer population in a Pennsylvania county between 2010 and 2014. Assume that the population grows exponentially. (a) What was the deer population in 2010? (b) Find a function that models the deer population t years after 2010. (Enter your answer in the form n0e rt. Round your r value to four decimal places (c) What is the projected deer population in 2024? (Round your answer to the nearest thousand.) (d) Estimate how long it takes the population to reach 400,000. (Round your answer to two decimal places.) The count in a culture of bacteria was 800 after 2 hours and 51,200 after 6 hours. (a) What is the relative rate of growth of the bacteria population? Express your answer as a percentage. (Round your answer to the nearest whole number.)
(b) What was the initial size of the culture? (Round your answer to the nearest whole number.) (c) Find a function that models the number of bacteria n(t) after t hours. (Enter your answer in the form n0e rt. Round your n0 value to the nearest whole number. Round your r value to two decimal places.) (d) Find the number of bacteria after 4.5 hours. (Round your answer to the nearest hundred.) (e) After how many hours will the number of bacteria reach 100,000? (Round your answer to two decimal places.) The half-life of radium-226 is 1600 years. Suppose we have a 24-mg sample. (a) Find a function m(t) = m02 t/h that models the mass remaining after t years. (b) Find a function m(t) = m0e rt that models the mass remaining after t years. (Round your rvalue to six decimal places.) (c) How much of the sample will remain after 5000 years? (Round your answer to one decimal place.) (d) After how many years will only 15 mg of the sample remain? (Round your answer to one decimal place.) The half-life of strontium-90 is 28 years. How long will it take a 100-mg sample to decay to a mass of52 mg? (Round your answer to the nearest whole number.) Radium-221 has a half-life of 30 sec. How long will it take for 96% of a sample to decay? (Round your answer to the nearest whole number.) After 3 days a sample of radon-222 has decayed to 58% of its original amount. (a) What is the half-life of radon-222? (Round your answer to two decimal places.) (b) How long will it take the sample to decay to 20% of its original amount? (Round your answer to two decimal places.) (a) Find the magnitude of an earthquake that has an intensity that is 74.7 (that is, the amplitude of the seismograph reading is 74.7 cm). (Round your answer to one decimal place.) (b) An earthquake was measured to have a magnitude of 5.9 on the Richter scale. Find the intensity of the earthquake. (Round your answer to one decimal place.) Earthquake A had a magnitude of 8.3 on the Richter scale. At the same time an earthquake B with magnitude 4.9 caused only minor damage. How many times more intense was earthquake A than earthquake B? (Round your answer to two decimal places.)
(a) The intensity of the sound from the speakers of a certain MP3 player (without earbuds) is measured at 3.2 10 5 W/m 2. Find the decibel level. (Round your answer to the nearest whole number.) (b) If earbuds are used with the MP3 player in part (a), the decibel level is 95 db. Find the intensity. (Round your answer to three decimal places.) (c) Find the ratio of the intensity of the sound from the MP3 player with earbuds to that of the sound without earbuds. (Round your answer to one decimal place.)