lim a, where and x is any real number. Exponential Function: Has the form y Graph y = 2 x Graph y = -2 x Graph y = Graph y = 2

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1 Precalculus Notes Da 1 Eponents and Logarithms Eponential Function: Has the form a, where and is an real number. Graph = 2 Graph = Graph = 2 Graph = The Natural Base e (Euler s number): An irrational number, smbolized b the letter e, appears as the base in man applied eponential functions. This irrational number is approimatel equal to More accuratel, The number e is called the natural base. The function is called the natural eponential function. lim 1 1 e

2 Evaluate: a. e b. e 3 c. e -2 Formulas for Compound Interest: After t ears, the balance A in an account with principal P and annual interest rate r (epressed as a decimal) is given b the following formulas. For n compoundings per ear: For continuous compounding: A P 1 rt A Pe \ r n nt Continuous is not offered but used to figure the maimum earnings at a given rate. E: The total invested is $1750 with an annual interest rate of 2% that is compounded earl. How much mone will be in the account after 7 ears? E: You have $10,000 to invest at 4% interest compounded quarterl or continuousl for 5 ears. What is the difference in the investments? E: Suppose our communit has 4512 students this ear. The student population is growing 2.5% each ear. E: Technetium-99 has a half-life of 6 hours. Suppose a lab has 80 mg of technetium-99. How much technetium-99 is left after 24 hours? E: The half-life of ioding-131 is 8 das. Suppose ou start with a 50-millicuries sample of iodine-131. How much iodine-131 is left after one half-life? After two half-lives?

3 Da 2 Definition of a Logarithmic Function: The logarithmic function log a, where a > 0 and a 1, is the inverse of the eponential function a. Logarithms to Eponential and vice versa: n = b p <=> p = log b n n = number b = base p = power E: Epress log 2 8 = 3 as an eponent E: Epress = as a log 81 Graphs of Eponential and Logarithm: Eponential has Logarithm has Horizontal Asmptote = 0 Vertical Asmptote = 0 Composites = 3 and log 3 Inverse Properties of Logarithms: For power. The logarithm with base b of b raised to a power equals that b raised to the logarithm with base b of a number equals that number. Evaluate each: a. b. c. d.

4 E: Evaluate log = E: Evaluate log 4 = 2 1 E: Evaluate log b 81= 2 E: Find the log = E: log 8 = 3 2 E: log 3 (4 + ) = log 3 (2) Find the domain of each logarithmic function: a. b. c. Solve for : E: log b (3 6) = log b ( + 4) E: log 2 (2 + 4) = log 2 ( 5) Properties: Eponential Logs 1. a m * a n = a m + n 1. logmn = logm + logn 2. a m a n = a m - n 2. log n m = logm logn 3. (a m ) n = a m*n 3. logm 3 = 3logm 4. a 1 = a 2 <=> 1 = 2 4. log 1 = log 2 <=> 1 = 2 If b > 0 and b 1 then log 1 = log b 2 <=> 1 = 2 *make sure 1 and 2 does not equal a negative. E: log 3 56 log 3 8 = log 3 E: 4log 10 = log E: log 10 ( ) = log E: log log 2 7 = log 2

5 Da 3 Review Properties: E: Epand log E: Condense 3 1 ((ln 2 + ln 6)) ln 4 Solve: E: 2 log log6 8 = log 6 E: 2log 2 log 2 ( + 3) = 2 Given log 2 5 = Find log 2 20 = log 2 25 = Given log 12 9 =.884 log = E: Find log 12 ( 4 3 ) = E: Find log = Common log f() = log 10 base of 10 Natural log f() = log e = ln base of e All properties of logarithms also hold for natural logarithms. Graph f() = log 2 ( 5) How do we graph logs with different bases? Change of Base Formula: log a n = log log n a E: log 4 22 = log log E: log =

6 Da 4 Eponential Equations: E: 4 = 24 E: 7 = 20 E: 7-2 = 5 3- E: e = 72 E: 4e 2 = 5 E: e 2 3e + 2 = 0 E: 5 + 2ln = 4 E: 2ln 3 = 4 E: A population of a town increases according to the model P(t) = 2500e.0293t with t = 0 corresponding to 1990 What will the population be in 2010? When will the population increase beond 8,000 people? E: The number of trees per acre N of a certain species is N(t) = 68(10-04 ) 5 40 where is the average diameter of trees 3 feet above the ground. Use the model to approimate the average diameter when N = 21.

7 Da 5 Eponential and Logarithmic Models 1. Eponential Growth and Deca Models 2. The table shows the growth of the minimum wage from 1970 through In, 1970, the minimum wage was $1.60 per hour. B 2000, it had grown to $5.15 per hour $1.60 $2.10 $3.10 $3.35 $3.80 $4.25 $5.15 a. Find the eponential growth function that models the date for 1970 through b. B which ear will the minimum wage reach $7.50 per hour? 3. In 1990, the population of Africa was 643 million and b 2000 it had grown to 813 million. a. Use the eponential growth model in which t is the number of ears after 1990, to find the eponential growth function that models the data. b. B which ear will Africa s population reach 2000 million, or two billion? 4. Use the fact that after 5715 ears a given amount of carbon-14 will have decaed to half the original amount to find the eponential deca model for carbon-14. a. In 1947, earthenware jars containing what are known as the Dead Sea Scrolls were found b an Arab Bedouin herdsman. Analsis indicated that the scroll wrappings contained 76% of their original carbon-14. Estimate the age of the Dead Sea Scrolls.

8 5. Gaussian Models This tpe of model is commonl used in probabilit and statistics to represent populations that are normall distributed. The graph of a Gaussian model is called a bell-shaped curve. 6. Logistic Growth Models This model grows rapidl then has a declining rate of growth (growth slows down). 7. The function describes the number of people, who have become ill with influenza t weeks after its initial outbreak in a town with 30,000 inhabitants. a. How man people became ill with the flu when the epidemic began? b. How man people were ill b the end of the fourth week? c. What is the limiting size of the population that becomes ill?

9 8. The table represents the federal minimum wages that were not adjusted for inflation from 1970 through 2000., Number of Years after 1969, Federal Minimum Wage a. Use the graphing calculator to find an eponential model for this data. b. Use the graphing calculator to find a logarithmic model for this data. c. Use the graphing calculator to find a linear model for this data. d. Use the graphing calculator to find a power model for this data. e. Which model fits the data the best? Use this model to predict which ear the minimum wage will reach $7.50. How does this answer compare to the answer we found in number 1?

The Exponential function f with base b is f (x) = b x where b > 0, b 1, x a real number

Chapter 4: 4.1: Exponential Functions Definition: Graphs of y = b x Exponential and Logarithmic Functions The Exponential function f with base b is f (x) = b x where b > 0, b 1, x a real number Graph: