EXPONENTS AND LOGS (CHAPTER 10)

Transcription

1 EXPONENTS AND LOGS (CHAPTER 0)

2 POINT SLOPE FORMULA The point slope formula is: y y m( ) where, y are the coordinates of a point on the line and m is the slope of the line. ) Write the equation of a line that passes through the point (-, ) and has a slope of. ) Write the equation of the line that passes through the points (-, ) and (-, 5). ) A sales person is paid a daily salary plus commission. When his sales are $000, he makes $00. When his sales are $00, he makes $0. Write a linear equation to model this situation. ) Write an equation for the line that passes through (-, ) and is perpendicular to the line whose equation is y = -. 5) Write an equation using the point slope formula for each graph.

3 FRACTIONAL EXPONENTS n a Re-write using fractional eponents: ) 5 ) ) ) 5) y 6) y 7) 5 b 8) Re-write the epression using a radical sign: 9) 0 5 0) ) 6 5 ) ) a = ) ab = 5) 5 6) 6 7) If f() =, find f(6). 8) Evaluate a 0 + a + a - when a = 8.

4 EVALUATING FRACTIONAL EXPONENTS BY HAND Without using your calculator evaluate the following: ) 8 ) 7 5 ) 5 6 ) 5) The value of 6 is a) -6 b) 6 c) 5 d) 6 6) Find the value of a) 8 7 b) 9 c) 9 d) 9 7) What is the value of if = 8? a) b) c) d) 8) What is the value of ( ) if = 7? a) b) c) d) -

5 SOLVING USING FRACTIONAL EXPONENTS ) = 8 ) y 6 w ) 9 ) 5 9 5) a ) z 7) 6 8) 8 5 9) 5 0) ) 5 ( ) 9 ) r 5 5

6 LOGARITHMS RECAP ) Graph the function y = log on the graph below and fill in the table. y a) What is the domain? b) What is the range? c) What is the limlog? d) What is the limlog? 0 6

7 CONVERTING AN EXPONENTIAL FUNCTION TO A LOGARITHMIC FUNCTION Write in log form: ) 8 = 6 ) 5 / = 5 Write in eponential form: ) log79 = ) log 8 Solve for : (Must rewrite B E = N logbn = E) 5) log = 6) log7 = 7 7) log6 = 8) log8 = 7

8 APPLICATIONS OF LOGS ) Lori and Ed were sightseeing in the desert when their camper ran out of gas along a level stretch of interstate highway. The speed of their camper decreased eponentially over time. The camper s speed function is represented by P(t) where the speed, P, is measured in miles per minute, and t is epressed in minutes. P(t) =.( ) t To the nearest minute, how long did it take until their speed was 0.0 mile per minute? ) It has been shown that homes in a certain city increase in value at a rate of 7.5% per year. The value, V, of a home after t years is given by the formula V = C( + r) t where r is the rate of appreciation. If a home costs $,000 in 00, during what year will this home have doubled in value? 8

9 ) The percentage of the US population that is foreign-born is growing at an eponential rate. The function is represented by the equation P(t) =.5907(.07) t where P is in millions and t is the number of years since 970. In what year did the number of people born outside the US double their population of 970? ) A super bouncy ball is dropped from a height of feet. Each time it bounces, it rises to a height of 80% of the height from which it fell. The height, h, can be determined by the equation h = (.80), where is the number of bounces. Determine the number of bounces necessary for the ball to be at most feet from the floor. 9

10 LOG PROPERTIES AND SOLVING Epand the following epressions using the properties of logs: ) log w c ) r ab 7 log ) log a (bc) c Re-write the following using a single log: ) log d log c + log e 5) loga logb logc Use log properties to solve the following: ) log9 + logn = log7 ) log70 log70 = log7n ) log log( ) = ) log ( - ) + log ( ) = 0

11 NATURAL EXPONENT For many applications, the convenient choice for a base is the irrational number e = This number is called the natural base. The function f() = function. Let s graph the function f() = e below. e is the natural eponential What is the domain of f() = e? What is the range of f() = e? What is the y-intercept? Fill in the following table by substituting the following values: X From this we can conclude that lim

12 One of the most familiar eamples of eponential growth is that of an investment earning continuously compounded interest. rt Our formula for this is A Pe, which gives us the balance A in an account with principal P and annual interest rate r, after t years. Eample: $9000 is invested at an annual interest rate of 9% compounded continuously. Find the balance after 5 years. ) The approimate number of fruit flies in an eperimental population after t hours is Q(t).0t 0e a) Find the initial number of fruit flies in the population. b) How large is the population of fruit flies after 7 hours? ) Let y represent the mass of a quantity of a radioactive element whose half-life is 5.0t years. After t years, the mass in grams is y = 0e. What percent of the present amount of the element will remain after 5 years?

13 THE NATURAL LOGARITHMIC FUNCTION The function defined by f() loge ln, where 0 is called the natural logarithmic function. Let s graph the function f() = ln below. What is the domain of f() = ln? What is the range of f() = ln? What is the - intercept? How does this graph relate to the graph of y = e?

14 PROPERTIES OF NATURAL LOGARITHMS ln = because ln e = because 0 e =. e = e. ln e = because e ln. If ln = ln y, then. ) Use the properties of natural logarithms to rewrite each epression. a) b) ln e ln e c) 0 ln e d) lne ) Find the domain and range of the following: a) f() = ln ( ) b) g() = ln ( ) c) h() = ln ( )

15 ) You deposited $000 in an account that pays.5% interest, compounded continuously. a) How much is in the account after 5 years? b) How long does it take for the money the double? ) From 970 to 008, the Consumer Price Inde (CPI) value y for a fied amount of sugar for the year t can be modeled by the equation y lnt where t = 0 represents 970. During which year did the price of sugar reach.5 times its970 price of 8.8 on the CPI? Solve for : ) ln ln 0 ) e ) ln ) ln( ) 0 5

16 5) ln 7 6) ln( 5) ln( ) ln( ) 7) The number of trees per acre N of a certain species is approimated by the model N 68(0.0 ) where is the average diameter of the trees, in inches, three feet above the ground. Use the model to approimate the average diameter of the trees in a test plot when N =. 8) On a college campus of 5000 students, one student returns from vacation with a contagious flu virus. The spread of the virus is modeled by 5000 y,wheret.8t 999e > 0 where y is the total number infected after t days. The college will cancel classes when 0% or more of the students are ill. a) How many students are infected after 5 days? b) If the outbreak continues, what would be the first day that the college cancels classes? 6