3.6.1 Classwork

A. Use a calculator to fill in the tables and sketch the two functions on the same graph.

1. [latex]f(x)=10^x[/latex]

   [latex]{\color[rgb]{1.0, 1.0, 1.0}\boldsymbol x}[/latex]	[latex]{\color[rgb]{1.0, 1.0, 1.0}\boldsymbol y}{\color[rgb]{1.0, 1.0, 1.0}\mathbf=}{\color[rgb]{1.0, 1.0, 1.0}\mathbf10}^{\color[rgb]{1.0, 1.0, 1.0}\mathbf x}[/latex]
   [latex]-3[/latex]
   [latex]-2[/latex]
   [latex]-1[/latex]
   [latex]0[/latex]
   [latex]1[/latex]
   [latex]2[/latex]

   

2. [latex]g(x)=\log(x)[/latex]
   

   [latex]{\color[rgb]{1.0, 1.0, 1.0}\boldsymbol x}[/latex]	[latex]{\color[rgb]{1.0, 1.0, 1.0}\boldsymbol y}{\color[rgb]{1.0, 1.0, 1.0}\mathbf=}{\color[rgb]{1.0, 1.0, 1.0}\boldsymbol l}{\color[rgb]{1.0, 1.0, 1.0}\boldsymbol o}{\color[rgb]{1.0, 1.0, 1.0}\boldsymbol g}\mathbf{\color[rgb]{1.0, 1.0, 1.0}\left(x\right)}[/latex]
   [latex]0.001[/latex]
   [latex]0.01[/latex]
   [latex]1[/latex]
   [latex]10[/latex]
   [latex]100[/latex]

3. How are the x- and y-coordinates of these two functions related? Give the domain and range for each function.

B. Use a calculator to evaluate. Give exact answers when possible, and round approximate answers to three decimal places.

4. [latex]\log(1)[/latex]
5. [latex]\log(8)[/latex]
6. [latex]\log(800)[/latex]
7. [latex]\log(10,000)[/latex]

8. 1. How are the inputs changing in problems 4 through 7 above?
   2. How are the outputs changing in problems 4 through 7 above?
   3. What is the difference between how the inputs are changing versus how the outputs are changing in problems 4 through 7 above?
   4. When will log give an integer answer?

C. Evaluate.

9. [latex]\sqrt{5^2}[/latex]
10. [latex]\log\left(10^9\right)[/latex]
11. [latex]\log\left(10^7\right)[/latex]
12. [latex]10^{\log\left(2\right)}[/latex]
13. [latex]10^{\log\left(5\right)}[/latex]

D. Solve.

14. [latex]400=10^x[/latex]
15. [latex]50,000=50{(10)}^x[/latex]
16. [latex]\log(2x)=0[/latex]
17. [latex]\log\left(x-2\right)=1[/latex]
18. [latex]10^x+8=20[/latex]
19. [latex]7=-\log\left(x\right)[/latex]

E. Solve. Round approximate answers to two decimal places (unless stated otherwise).

20. As of today, the population of the United States is approximately ______________ million people with an estimated exponential growth rate of ______________ % per year.
    1. What is the expected population (to the nearest million) of the U.S. this time in 2025?
    2. When will the US population hit half a billion?
    3. Why are these answers estimates?
    4. How would the answers above change if growth was linear instead of exponential?
21. Valium decays exponentially in the body according to the model [latex]V=V_0e^{kt}[/latex] where V is the amount of Valium left after an initial dose of [latex]V_0[/latex], with a decay rate of k over time t (in hours). If [latex]10[/latex] mg of Valium decays to [latex]5[/latex] mg in 36 hours (this is called the half-life), find the rate of decay of Valium and the sign of the rate of decay.
22. The pH scale measures how acidic or basic (alkaline) a substance is. It ranges from 0 to 14. A pH of 7 is neutral. A pH less than 7 is acidic. A pH greater than 7 is basic (alkaline). Use the formula [latex]pH=-\log\lbrack H^+\rbrack[/latex] where [latex]H^+[/latex] is the hydronium ion concentration. Root beer has a hydronium ion content of [latex]9.16\times10^4[/latex] mol/L. What is the [latex]pH[/latex]?
23. The [latex]pH[/latex] of ammonia is about [latex]11.5[/latex]. What is the hydronium content?
24. Determine the [latex]pH[/latex] of lemon juice which has a hydronium ion content of [latex]3.98\times10^{-3}[/latex] mol/L.
25. An acid reflux diet requires foods and drinks with a [latex]pH[/latex] higher than [latex]5[/latex]. Would root beer or lemon juice be allowable for this diet? Explain.
26. Using the formula [latex]R=\log\left(\frac x{0.001}\right)[/latex], where x is the seismographic reading in millimeters, and R is the magnitude of the earthquake on the Richter scale, determine the magnitude of an earthquake with a seismographic reading of [latex]23,750[/latex] mm.
27. What was the seismograph reading for the earthquake on the coast of Japan in 2011, which measured a magnitude of [latex]9.1[/latex]?

3.6.2 Homework

A. Solve.

1. [latex]10^x=30[/latex]
2. [latex]10^x-4=96[/latex]
3. [latex]3{(10)}^x=9[/latex]
4. [latex]\log(x-2)=2[/latex]
5. [latex]\log(3x)=0[/latex]
6. [latex]6000=P\cdot10^{0.02\cdot10}[/latex]
7. [latex]6000=30\cdot10^{0.05\cdot x}[/latex]
8. [latex]8\left(10^{3x}\right)=12[/latex]

B. Solve. Round approximate answers to two decimal places.

9. In the 2010 Census, the population of San Antonio was 1.334 million people. In 2013, the population rose to an estimated 1.39 million.
   1. What was the exponential growth rate of San Antonio during this period?
   2. Use the growth rate found in part (a) to estimate the population of San Antonio in 2030.
   3. Estimate when San Antonio will hit 2 million people.
10. If the number of telephone solicitations increases exponentially (continuously), how long will it take to double your number of annoying phone calls if the rate of growth is [latex]12%[/latex] per day?
11. A certain bacteria population can be modeled by the function [latex]N=25e^{kt}[/latex] where [latex]N[/latex] is the number of bacteria (in thousands) after [latex]t[/latex] minutes, and [latex]k[/latex] is the rate of growth. After [latex]4[/latex] minutes, there are [latex]100,000[/latex] bacteria. Find the rate of growth.
12. Certain radioactive decay can be modeled by the function [latex]R=e^{kt}[/latex] where R is the ratio of remaining material, k is the rate of decay, and t is time in years. When 1,000 years have passed, [latex]R=0.65[/latex]. (This means that after 1,000 years, the material still has [latex]65%[/latex] of its radioactive energy.) Find the rate of decay. What is its sign and why?
13. The metabolism of Ibuprofen in a patient’s system can be modeled by [latex]V=V_0e^{kt}[/latex], where [latex]V_0[/latex] is the initial dosage and V is the amount of Ibuprofen remaining after t hours. If the decay rate of Ibuprofen is approximately [latex]-34.65\%[/latex] per hour, what is its half-life?

14. Determine the hydronium ion concentration of human blood, which has [latex]pH=7.4[/latex].
15. If soil has an [latex]H^+[/latex] concentration of  [latex]1.995\times10^{-5}[/latex] mol/L, what is its [latex]pH[/latex]?
16. Find the hydronium ion concentration of orange juice [latex](pH=3.5)[/latex].
17. Lime water has a hydronium ion content of [latex]3.98\times10^{-13}[/latex]. Find its pH.

18. Determine the magnitude of an earthquake with a seismographic reading of [latex]12,298[/latex] mm.
19. There was a small, [latex]4.8[/latex] magnitude earthquake on IH-37 near Pleasanton in November 2011. Find its seismographic reading (in mm).
20. Find the seismographic reading (in mm) of the Virginia earthquake from August 2011. It had a Richter scale magnitude of [latex]5.8[/latex].
21. How many times stronger (in seismographic reading) is a [latex]5.5[/latex] Richter scale earthquake than a [latex]3.5[/latex] Richter scale earthquake?
22. Determine the Richter scale magnitudes of earthquakes with seismographic readings of [latex]5,000[/latex] mm and [latex]50,000,000[/latex] mm. What did you notice?