Chapter 6A Solving Exponential and Logarithmic Equations. Solve x+5 = x = 9 x x 2 = x 4. 5 x = 18

Transcription

1 Fry Texas A&M University!! Math 150!! Chapter 6!! Fall 2014! 1 Chapter 6A Solving Exponential and Logarithmic Equations Solve x+5 = x = 9 x x 2 = x 4. 5 x = 18

2 Fry Texas A&M University!! Math 150!! Chapter 6!! Fall 2014! e 1 x = 1 6. e 2 x 2e x = 3

3 Fry Texas A&M University!! Math 150!! Chapter 6!! Fall 2014! 3 7. log 5 x = 3 8. log 7 x = 2 9. log 1 x = log 2 (3x + 5) = log x + log(x 21) = 2

4 Fry Texas A&M University!! Math 150!! Chapter 6!! Fall 2014! log 2 (x 2 6x) = 3+ log 2 (1 x) Suggested Problems: Text: 1-16 My Previous Exams:!! S14 3A: 1e, 3d!! F13 3A: 8!!!!!!! S13 3A: 5!!! F12 3A: 4, 7 Dr. Scarborough s Previous Exams: F13 3: p7: 6, 7 Dr. Scarborough s Fall 2013! WIR 7: 7, 16!!!!!! WIR 8: 4, 9, 19!!!!!! WIR 9: 18, 42, 45, 50, 52, 56

5 Fry Texas A&M University!! Math 150!! Chapter 6!! Fall 2014! 5 Dr. Kim s Fall 2014 WIR: Chapter 6B - Applications of Exponentials and Logarithms 1. A biologist has a 100 bacteria in a petri dish. She measures their growth over many hours and finds that the population doubles every hour. Complete the table below. Determine the pattern. Write an equation that models the population at any time t. t in hours Population Repeat the strategy above using these conditions. The initial population of bacteria is 10. The population increases by a factor of 5 every 3 hours. t in hours Population

6 Fry Texas A&M University!! Math 150!! Chapter 6!! Fall 2014! 6 3. Jason started this great new Facebook page and invited his friends to Like it. It started with just 1 Like, but every 15 minutes the number of Likes tripled. Complete the table below to help establish a pattern. Then write a function that describes the number of Likes at time t. The populations above are said to follow an exponential growth law P(t) = P(0)a kt P(0) is In some cases, you may see this denoted P 0. It is read P-zero or sometimes P naught. a and k are determined by the setting. Sometimes it is not necessary to know a and k as individuals. Sometimes it is enough to know a k. In fact, since a = e lna, then ( a) kt = e lna ( ) kt = e (lna)kt = e k* t So most people solve exponential growth problems with the basic formula P(t) = P(0)e kt 4. Suppose P(t) satisfies an exponential growth law. If P(0) = 50 and P(3) = 400, find P(5).

7 Fry Texas A&M University!! Math 150!! Chapter 6!! Fall 2014! 7 5. During its exponential growth phase, a certain bacterium can grow from 5,000 cells to 12,000 cells in 10 hours. a) At this rate, how many cells will be present after 36 hours? b) How long will it take to grow to 50,000 cells.

8 Fry Texas A&M University!! Math 150!! Chapter 6!! Fall 2014! 8 6. The population of a certain city in 1975 was 65,000. In 2000 the census determined that the population was 99,500. Assuming exponential growth, estimate the population in 2015.

9 Fry Texas A&M University!! Math 150!! Chapter 6!! Fall 2014! 9 7. The half-life of a radioactive substance is the amount of time it takes for one-half of the original amount of the substance to change into something else. That is, after each half-life, the amount of the original substance decreases by one-half. In 1898 Marie Curie discovered the highly radioactive element radium. She shared the 1903 Nobel Prize in physics for her research on radioactivity and was awarded the 1911 Nobel Prize in chemistry for discovery of radium and polonium. Radium 226 (an isotope of radium) has a half-life of 1601 years and decays into radon gas. In a sample originally having 1 gram of radium 226, the amount of radium 226 present after t years is given by A(t) = 1 t How much radium 226 will be present after 3202 years? Half-life Formula: Given an initial amount of material A 0 with a half life of h, the amount of material at a given time t can be determined by the following model: 1 A(t) = A 0 2 t h Take the time to consider how this model makes sense for t = 0, t = h, t = 2h, etc. Some people prefer to use models involving e. It is not difficult to show that the following model is equivalent to the one above:!!!!! A(t) = A 0 e t ln 2 h

10 Fry Texas A&M University!! Math 150!! Chapter 6!! Fall 2014! Plants fix atmospheric carbon during photosynthesis, so the level of Carbon-14 in plants and animals when they die approximately equals the level of Carbon-14 in the atmosphere at that time. However, it decreases thereafter from radioactive decay, allowing the date of death or fixation to be estimated. Given that the half-life of carbon-14 is 5730 years, estimate the age of a skeleton that has 95% of its Carbon-14 remaining.

11 Fry Texas A&M University!! Math 150!! Chapter 6!! Fall 2014! 11 Suggested Problems: Text: 1-15 (On #15, ignore the - sign after the 40. It is a 40 g sample.) My Previous Exams:!! S14 3A: 8, 9! F13 3A: 7,!!!!! S13 3A: 7,! F12 3A: 8 Dr. Scarborough s Previous Exams: F13 3: p6: 4 Dr. Scarborough s Fall 2013! WIR 8: 2, 5, 6, 7, 11, 12, 17!!!!!! WIR 9: 36, 55 Dr. Kim s Fall 2014 WIR: