3.4E: Solving Exponential Equations with Logarithmic Functions (Exercises)
section 3.4 exercises
Part 1: Using logarithmic functions and their properties to solve exponential equations
1. Rewrite each equation in exponential form
a. \(\log _{4} (q)=m\)
b. \(\log _{3} (t)=k\)
c. \(\log _{a} (b)=c\)
d. \(\log _{p} (z)=u\)
e. \(\log \left(v\right)=t\)
f. \(\ln \left(w\right)=n\)
2. Rewrite each equation in logarithmic form.
a. \(4^{x} =y\)
b. \(5^{y} =x\)
c. \(c^{d} =k\)
d. \(n^{z} =L\)
e. \(10^{a} =b\)
f. \(e^{k} =h\)
3. Evaluate each expression using your calculator to find a decimal approximation.
a. \(\log \left(0.04\right)\)
b. \(\log \left(1045\right)\)
c. \(\ln \left(15\right)\)
d. \(\ln \left(0.02\right)\)
4. Find a decimal approximation of each logarithm using the change of base formula and a calculator.
a. \(\log _{4} (5)\)
b. \(\log _{8} (12)\)
c. \(\log _{3} (\dfrac{1}{15})\)
In exercises 5 - 22, solve each equation for the unknown variable. Then write your solution with a decimal approximation.
5. \(5^{x} =14\)
6. \(3^{x} =23\)
7. \(7^{x} =\dfrac{1}{15}\)
8. \(e^{5x} =17\)
9. \(3^{4x-5} =38\)
10. \(1000\left(1.03\right)^{t} =5000\)
11. \(200\left(1.06\right)^{t} =550\)
12. \(3\left(1.04\right)^{3t} =8\)
13. \(2\left(1.08\right)^{4t} =7\)
14. \(50e^{-0.12t} =10\)
15. \(10e^{-0.03t} =4\)
16. \(10-8\left(\dfrac{1}{2} \right)^{x} =5\)
17. \(\log _{3} \left(x\right)=2\)
18. \(\log _{4} (x)=3\)
19. \(5\log _{2} (x)+4=-3\)
20. \(3\log _{5} (x)=-1\)
21. \(\log \left(x\right)=3\)
22. \(\ln \left(x\right)=2\)
Part 2: Using the properties of logarithms
In exercises 23 - 32, simplify each expression using properties of logarithms. In some cases, it will be helpful to write the input of the logarithm as a power of the base. In other cases, this may not be easily done.
23. \(\log _{5} \left(25\right)\)
24. \(\log _{2} \left(8\right)\)
25. \(\log _{3} \left(\dfrac{1}{27} \right)\)
26. \(\log _{6} \left(7^{3}) \right)\)
27. \(\log _{6} \left(\sqrt{6} \right)\)
28. \(\log _{5} \left(\sqrt[{3}]{5} \right)\)
29. \(\log \left(10,000\right)\)
30. \(\log \left(0.001\right)\)
31. \(\log \left(5^{20}\right)\)
32. \(\ln \left(e^{3} \right)\)
Part 3: More on Modeling with exponential functions
33. You take 200 milligrams of a headache medicine, and after 4 hours, 120 milligrams remain in your system.
a. Find a continuous exponential formula that models the milligrams of the medicine remaining in your system after t hours.
b. If the effects of the medicine wear off when less than 80 milligrams remain, when will you need to take a second dose, assuming the amount of medicine in your system decays exponentially?
34. You go to the doctor and they inject you with 13 milligrams of radioactive dye as part of a test. After 12 minutes, 4.75 milligrams of dye remain in your system.
a. Find a continuous exponential formula that models The amount of dye in your system after t minutes.
b. To leave the doctor’s office, you must pass through a radiation detector without sounding the alarm. If the detector will sound the alarm whenever more than 2 milligrams of the dye are in your system, how long will your visit to the doctor take, assuming you were given the dye as soon as you arrived and the amount of dye decays exponentially.
35. A ship embarked on a long voyage. At the start of the voyage, there were 500 ants in the cargo hold of the ship. One week into the voyage, there were 800 ants. Suppose the population of ants is an exponential function of time. [UW]
a. Find a continuous exponential formula for the number of ants, A, after t weeks.
b. How long did it take the population to double?
c. When were there be 10,000 ants on board?
36. The population of Seattle grew from 563,374 in 2000 to 608,660 in 2010. Assume the population continues to grow exponentially at the same rate.
a. Find a continuous exponential model for P, the population of Seattle, t years since 2010.
b. When will the population exceed 1 million people?
37. The median household income (adjusted for inflation) in Seattle grew from $42,948 in 1990 to $45,736 in 2000. Assume it continues to grow exponentially at the same rate.
a. Find a continuous exponential model for I, the median household income in Seattle, t years since 1990.
b. When will median income exceed $70,000?
38. A radioactive substance decays exponentially. A scientist begins with 110 milligrams of a radioactive substance. After 31 hours, 55 mg of the substance remains.
a) Find a continuous exponential formula for the amount of the radioactive
b) How long until only 10 mg remain?
39. The population of Kenya was 39.8 million in 2009 and has been growing by about 2.6% each year. Assume this trend continues
a. Find a formula for P, the population of Kenya t years since 2009.
b. When will the population exceed 60 million?
40. The number of firearm homicide deaths was 4.6 deaths per 100,000 people in 2019. The number of firearm homicide deaths increased by 35% per year from 2019 to 2020. 1 Assume this trend continues.
a. Find a formula for N, the number of firearm homicide deaths, t years since 2019.
b. Predict when the number of firearm homicide deaths will reach 10 deaths per 100,000 people.
41. A bacteria culture initially contains 1500 bacteria and doubles in size every half hour.
a. Find a formula for N, the number of bacteria, in t hours.
b. How long until there are 10,000 bacteria?
42. The half-life of Radium-226 is 1590 years. A sample initially contains 200 mg.
a. Find an exponential formula for the amount in t years.
b. How long until only 10 mg remain?
43. The half-life of Fermium-253 is 3 days. A sample initially contains 100 mg.
a. Find a formula for the amount in t years.
b. how many milligrams will remain after 1 week?
44. A scientist begins with 250 grams of a radioactive substance. After 225 minutes, the sample has decayed to 32 grams. Find the half-life of this substance.
45. A wooden artifact from an archeological dig contains 60 percent of the carbon-14 that is present in living trees. How long ago was the artifact made? (The half-life of carbon-14 is 5730 years.)
46. The count of bacteria in a culture was identified to be 800 after 10 minutes and 1800 after 40 minutes.
a. What was the initial size of the culture?
b. Find a formula for N, the number of bacteria, in t minutes.
c. Find the number of bacteria after 105 minutes.
d. When will the number of bacteria reach 11000?
47. Find the time required for an investment to double in value if invested in an account paying 3% compounded quarterly.
48. A turkey is pulled from the oven when the internal temperature is \(165^{\circ}\) Fahrenheit, and is allowed to cool in a \(75^{\circ}\) room. If the temperature of the turkey is \(145^{\circ}\) after half an hour,
a. What will the temperature be after 50 minutes?
b. How long will it take the turkey to cool to 110\(\mathrm{{}^\circ}\)?
49. A cup of coffee is poured at \(190^{\circ}\) Fahrenheit, and is allowed to cool in a \(70^{\circ}\) room. If the temperature of the coffee is \(170^{\circ}\) after half an hour,
a. What will the temperature be after 70 minutes?
b. How long will it take the coffee to cool to \(120^{\circ}\)\)?
50. As light from the surface penetrates water, its intensity is diminished. In the clear waters of the Caribbean, the intensity is decreased by 15 percent for every 3 meters of depth. Thus, the intensity will have the form of a general exponential function. [UW]
a. If the intensity of light at the water’s surface is\(I_{0}\), find a formula for \(I(d)\), the intensity of light at a depth of \(d\) meters. Your formula should depend on \(I_{0}\)and \(d\).
b. At what depth will the light intensity be decreased to 1% of its surface intensity?
51. The population of fish in a farm-stocked lake after t years could be modeled by the equation \(P\left(t\right)=\dfrac{1000}{1+9e^{-0.6t} }\).
a. Sketch a graph of this equation.
b. What is the initial population of fish?
c. What will the population be after 2 years?
d. How long will it take for the population to reach 900?
_____________________________________________________________________________________
1 - The data was reported by the CDC at https://www.cdc.gov/vitalsigns/firea...ths/index.html on February 9, 2023.
- Answer
-
Part 1:
1. a. \(4^m = q\) c. \(a^c = b\) e. \(10^t = v\)
2. a. \(\text{log}_{4} (y) = x\) c. \(\text{log}_{c} (k) = d\) e. \(\text{ln} (h) = k\)
4. a. \( \approx 1.161 \)
5. \(\dfrac{\text{ln}(14)}{\text{ln}(5)} \approx 1.6397\)
7. \(\dfrac{\text{log}(\dfrac{1}{15})}{\text{log}(7)} \approx -1.392\)
9. \(\dfrac{\dfrac{\text{log}(38)}{\text{log}(3)} + 5}{4} \approx 2.078\)
11. \(\dfrac{\text{ln}(2.75)}{\text{ln}(1.06)} \approx 17.361\)
13. \(\dfrac{\text{ln}(\dfrac{7}{2})}{\text{4ln}(1.08)} \approx 4.069\)
15. \(\dfrac{\text{ln}(0.4)}{-0.03} \approx 30.543\)
17. 919. \(2^{\dfrac{-7}{5}} \approx 0.3789\)
21. 1,000
Part 2:
23. 2
25. -3
27. \(\dfrac{1}{2}\)
29. 4
31. \(20\text{log} (5) \)
Part 3:
33. a. \(A = 200e^{-0.1277t}\) b. You will need to take a second dose after 7.18 hours.
35. a. \(A = 500e^{0.47t}\) b. The population will double in 1.47 days. c. In 6.37 days.
37. a. \(I = 42,948e^{0.00629t}\) b. The median income will exceed this level in 77.7 years.
39. a. \(P = 3.9(1.026)^{t}\) b. In 16.78 years.
41. a. \(N = 1500e^{1.386t}\) b. In 1.369 hours.
43. a. \(A = 100e^{-0.231t}\) b. 19.85 mg
45. It was made 4,222.7 years ago.
47. It will double in value in 23.19 years.
49. a. 148.4 degrees b. In 144 minutes.
51. a.
b. 100 fish c. 269 fish d. 7.324 years