Unit 9 Packet - Exponential And Logarithmic Functions
Name DatePre-Calculus 19-1 Properties of ExponentsPropertiesExampleam an x5 y 2 x10 y 3 (a )mn4 (x ) ( 2x y )m( ab )a m 3 25 x 4 y z 11 x 2 12 23 x 5 y 6 2 x17 y 2am an3m a b 3x 2 y a0 10( 2x y )24 Example 1: Simplify each of the following.(3a. 2 x3 y 2 3xy 7b. 2x3 y 2)2x3 y 2c.x5 z 4x3 y 2 d. 4 5 z y 21
e. 2 x1 2 y 4 x5 2 y 2(g. 2x1 2f. 1 2 10 x y z 4 4)All radicals can be written as exponents:y3 5y4 50h. nam am nExample 2: Write each of the following radicals as exponents.a.c.3xb.3xx4d.3x2Example 3: Write each of the following as rational exponents. Then, simplify without using a calculator.a.327 2b.4163c.3 8d.48132
Example 4: Simplify each of the following.12 4x8 y 6 a. 2 z Example 5: Rewrite27 2 xwith one base and no fraction.33 xExample 6: Rewrite125 xwith one base and no fraction.252 x(b. 27x9 y 1513)3
Name DatePre-Calculus 19-1 Properties of Exponents – Homework1. Simplify each expression.(a. a 2 3b3 2(6)34 32 23c. x35(re. 1 1 2 3y zs)rs 1 212 x 2 3 g. 3 4 y 5()b. 2x1 5 y 3 5)a 5 6b1 2d. 1 3 3 4a b13 8x3 y 6 f. 9 z 1m x3m y 2 m h. 5m z 4
13(i. 3x2 1 h2 3 27 x )( 4x ) 2 x 1 y 2 z 3 k. 2 0 x y z j. 1(l. 81x16 y 4 z 1214)227 x2. Which expression is equivalent to x :3a. 3x2 b. 3x 3 xc. 33 xd. 33 x2 x14 1512 13(( x ) ) 3. Simplify: 2 22a 3 8 , find the values of a and b. 3b 4 3 4. If 5
Name DatePre-Calculus 19-1 Properties of Exponents – Practice1. Simplify each of the following.(a. 2a 3 4b5a)bc. x x x(re.c 3 1 2 2ss 1 2)4(b. 2 x1 4 y 3 44) (3x 4y3 )216a 6b1 2d.2a 7 b 5 41312 8 x3 y 6 4 x6 f. 4 9 z y 6
(3g. 3x y 4 1 x3m y 2 m h. 5m z 3) ( 3x )30 m12(i. 2 x5 16 j. 6 x 2 x )( 4x ) x 1 y 2 z 3 k. 2 0 x y z 20(l. 16x16 y8 z 1214)2162 x2. Rewritewith one base and no fraction.4x7
Name DatePre-Calculus 19-2 Exponential FunctionsxExponential Function: For any real number x, an exponential function is a function in the form f ( x ) a b .There are two types of exponential functions:Exponential Growth:f ( x ) a b x , where b 1Exponential Decay:f ( x ) a b x , where 0 b 1a is the initial amount or starting value of the functionb is the growth or decay factorExample 1: In the year 2013, the population of India was about 1.25 billion, with an annual growth rate of 1.2%. This situationtis represented by the function p (t ) 1.25 (1.012 ) , where t is the number of years since 2013. What will the population ofIndia be in 2031?Example 2: In 2006, 80 deer were brought to a wildlife refuge. By 2012, the population had grown to 180 deer. The populationwas growing exponentially. Write an algebraic function to represent the number of deer over time.8
Alternate ways of writing exponential functions:Exponential Growth:y a(1 r ) xExponential Decay:y a(1 r ) xa is the initial amountr is the percent increase/decrease (expressed as a decimal)(1 r) or (1-r) is the growth factor or the decay factorExample 3: The value of the car, y (in thousands), can be approximated by the model y 25(0.85)t where t is the number ofyears since the car was new.a. Tell whether the model represents growth or decay.b. Identify the annual percent increase or decrease in the value of the car.c. Interpret the meaning of the 25.d. What will the value of the car be after ten years?Example 4: In 2000, the world population was about 6.09 billion. During the next 13 years, the world population increased byabout 1.18% each year.a. Write an exponential model given the population y (in billions), t years after 2000. Estimate the worldpopulation in 2005.b. What will be the population in the year 2020?9
The Nature Base e:Euler was a very smart man who has done a lot for the math world. He discovered the number that we are going to learnx 1 about. This number is based on the expression 1 as x .x It is an irrational number approximately equal to 2.718. Since e is just a number, we can use it as the base of an exponentialfunction.Exponential Growth:f ( x ) a ertExponential Decay:f ( x ) a ertExample 5: Radon-222 decays at a continuous rate of 17.3% each day. How much will 100 mg of Radon-222 decay in 3 days?Compound Interest:When interest is accumulated on the original value and on the accumulated value of the account. There are two compoundinterest formulas:Compounded Periodically: r A P 1 n ntA is the amount in the accountt is time in yearsP is the starting amount in the account, often called the principler is the annual percentage rate (APR), expressed as a decimaln is the number of compounding periods in one yearCompounded Continuously:A P ertA is the amount in the accountt is time in yearsr is the annual percentage rate (APR), expressed as a decimale is just a number approximately equal to 2.71810
Example 6: You deposit 9,000 in an account that pays 1.46% annual interest. Find the balance after three years when theinterest is compounded:a. Quarterlyb. Dailyc. ContinuouslyExample 7: Suppose you want to have 10,000 saved in an account after 25 years. If the bank account earns 2.1% annualinterest, how much money would you need to initially deposit in the bank if the interest is compounded:a. Weeklyb. ContinuouslyExample 8: You and your friend each have accounts that earn annual interest compoundedcontinuously. The balance A (in dollars) of your account after t years can be modeled by theequation A 4500e0.04t . The graph shows the balance of your friend’s account over time.a. Which account has greater principal?b. Which has a greater balance after 10 years?11
Name DatePre-Calculus 19-2 Exponential Functions – Homeworkt1. For each year t, the population for a forest of trees is represented by the function A (t ) 115 (1.025) . In a neighboringtforest, the population of the same type of tree is presented by the function B (t ) 82 (1.029 ) .a. Which forest’s population is growing at a faster rate?b. Which forest had a greater number of trees initially?c. Assuming the population growth models continue to represent the growth of the forests, which forest willgreater number of trees after 20 years?have ad. Assuming the population growth models continue to represent the growth of the forests, which forest willgreater number of trees after 100 years?have a 2. Use A ( t ) 10, 250 1 1200.04 12 to answer the following questions.a. What is the value of the account?b. What is the initial deposit?c. How many years has the account been accumulating interest?12
3. An account is opened with an initial deposit of 6,500 and earns 3.6% interest compounded semi-annually ( n 2 ).a. What will the account be worth in 20 years?b. How much more would be in the account if it were compounded weekly for 20 years?4. Calculate the initial deposit of an account that is worth 14,472.74 after earning 7.7% interest compounded monthly for 5years.5. The fox population in a certain region has an annual growth rate of 9% per year. In the year 2012, there were 23,900 foxcounted in the area. What is the fox population predicted to be in the year 2020?6. In the year 1985, a house was valued at 1100,000. By the year 2005, the value had appreciated to 145,000. What was theannual growth rate between 1985 and 2005? Assume that the value continued to grow by the same percentage.13
7. An investment account with an annual interest rate of 7% was opened with an initial deposit of 4,000. How much will bein the account after 9 years if the interest is compounded continuously?8. You take a 325 milligram dosage of ibuprofen. During each subsequent hour, the amount of medication in your bloodstreamdecreases by 29% each hour.a. Write an exponential model giving the amount y (in milligrams) of ibuprofen in your bloodstream t hours afterthe initial dose.b. Estimate how long it takes for you to have 100 milligrams of ibuprofen in your bloodstream.t9. The value of a mountain bike y ( in dollars) can be approximated by the model y 200 ( 0.75) , where t is the number ofyears since the bike was new.a. Tell whether the model represents exponential growth or decay.b. Identify the annual percent increase or decrease in the value of the bike.c. Estimate the value of the bike after 5 years.14
Name DatePre-Calculus 19-3 Graphing Exponential FunctionsTypically, we will see exponential functions in the formf ( x ) a b x h k . Let’s figure out what a, h, and k do to thefunction.Exploration: Let’s figure out what a, h, and k represent. Once you figure it out, go back and fill in what each variablerepresents on the previous page.a. Graphf ( x ) 3x on your calculator.b. Graph g( x ) 2 3x on the same axes as f ( x ) . Graph h ( x ) 1 3x on the same axes as f ( x ) . Graph2xt ( x ) 1 3 on the same axes as f ( x ) . What did each value of a do to the graph?c. Graph k( x ) 3x 2 and w ( x ) 3x 2 on the same axes as f ( x ) . What did these values of h do to the graph?d. Graph d( x ) 3x 2 and c ( x ) 3x 2 on the same axes as f ( x ) . What did these values of k do to thee. Graphgraph?z ( x ) 3 x on the same axes as f ( x ) . What did the negative do to the graph?b is the base of the exponentiala representsh representsk representsa negative in front of the x represents aThe starting point for an exponential function is:15
Asymptote: a line that a function approaches but never passes.Exponential functions will have an asymptote at:Example 1: Use transformations to graph each of the following functions.a.m ( x ) 3x 2 2b. t ( x ) ( 2 )x 4 3Equation of Asymptote:Equation of Asymptote:Domain:Domain:Range:Range: 1 2 c. c ( x ) x 1 3 1 3 xd f ( x ) 2Equation of Asymptote:Equation of Asymptote:Domain:Domain:Range:Range:16
e.f ( x ) e x 1 3 1 2 f. h ( x ) x 2 1Equation of Asymptote:Equation of Asymptote:Domain:Domain:Range:Range:17
Name DatePre-Calculus 19-3 Graphing Exponential Functions – HomeworkDirections: Sketch the graph of each of the following functions. Include one accurately plotted point and label the asymptote.Then, state the domain and range.1. 1 3 f ( x ) 2 x 2 3x 12. g ( x ) 23.h ( x ) 3 x 2Domain:Domain:Domain:Range:Range:Range:Equation of Asymptote:Equation of Asymptote:Equation of Asymptote: 3 2 4. t ( x ) x 3 1 1 2 5. j ( x ) x 4 2 1 2 x 26. z ( x ) Domain:Domain:Domain:Range:Range:Range:Equation of Asymptote:Equation of Asymptote:Equation of Asymptote:18
7.w ( x ) ex 48.c ( x ) 3x 4 19.z ( x ) 3x 1 1Domain:Domain:Domain:Range:Range:Range:Equation of Asymptote:Equation of Asymptote:Equation of Asymptote:10.r ( x ) ex 1 511.p ( x) 2x 2 4 1 12. a ( x ) 4 x 3 4Domain:Domain:Domain:Range:Range:Range:Equation of Asymptote:Equation of Asymptote:Equation of Asymptote:19
Name DatePre-Calculus 19-4 Logarithmic FunctionsWarm Up: Solve 2 x 6 . Can you think of any number off the top of your head (without using a calculator) that you couldxplug in for “x” what would make 2 equal to 6?You are probably struggling with coming up with an answer to the Warm Up. This is where logarithms come into play.Logarithms are functions that help us evaluate exponentials. Exponentials and logarithms are inverses of each other.Before we learn how to solve the Warm Up, we need to learn how to navigate between two different forms.logarithmic Form à logb y xb x y ß exponential formif and only ifRead: “Log base b of y equals x”Read: “b to the x equals y”Example 1: Rewrite each of the following in logarithmic form.a. 24 16b. 40 1c. 121 12d. 1 1 4 4 Example 2: Rewrite each of the following in exponential form.a. log5 25 2c. log 61 3216b. log8 4 d. log 623( 6 ) 1220
Evaluating Logarithms:Example 3: If I asked you to evaluatelog 2 (8) , what this is really asking is what exponent must 2 be raised in order to get an8? What is the answer?Example 4; Evaluate each of the following logarithms.a.log7 ( 49 )b. 1 27 log3 ( 27 ) 1 32 c. log 3 d. log 2 The Common Logarithm:Sometimes we will see a logarithm written without a base, such as log100 . In this case, we assume that the base is 10, solog100 log10 100 .Example 5: Evaluate each of the following logarithms.a. log100b. log1000Evaluating Logarithms on your Calculator:Ti-Nspire – Hit “ctrl” then “ 10 x ” Type in the log that you want to evaluateTi-83/84 – Your calculator is set as base 10. In order to evaluate, you must use the change of base formulaSay you wanted to find log2 4 , then log 2 4 log 4.log 2Now that we know how to go between forms and evaluate logarithms on our calculator. Let’s take a look at the warm up.21
Example 6: Solve 2 x 6 by rewriting in logarithmic form. Use your calculator for the last step.Example 7: The amount of energy released from an earthquake was 500 times greater than the amount of energy releasedfrom another. The equation 10 x 500 represents this situation where x is the difference in magnitudes on the Richter Scale.What was the difference in magnitudes?The Natural Logarithm:Since we learned that e is a number approximately equal to 2.718, it can also be the base of a logarithm. When a log has a basee we call this the natural logarithm. We write this a little bit differently. Instead of writing log e ( x ) , we write ln ( x ) .Example 8: Use your calculator to evaluateln (500) .22
Name DatePre-Calculus 19-4 Logarithmic Functions – Homework1. How are logarithmic functions and exponential functions related to each other?2. Rewrite each of the following in exponential form.a. log4 q mb. log a b cc. log16 y xd. log15 a be. log y 137 xf. log13 142 aln ( w) ni. log y x 11g.log ( v ) th.3. Rewrite each of the following in logarithmic form.a. 4 x yb. c d kc. m 7 nxkc. e h 7 d. y 5 e. 10a b23
3. Convert each exponential equation to logarithmic form. Then, solve for x.a.log3 ( x ) 2b.ln ( x ) 212d.log6 ( x ) 3b.log6 6c. log 9 ( x ) 4. Evaluate each of the following without using a calculator. 1 27 a. log 3 c.log (10,000)d. log5 1255. Use your calculator to evaluate the following logarithms.a.log ( 0.04 )b. ln1524
Name DatePre-Calculus 19-5 Properties of LogarithmsImportant Properties of Logarithms:1. logb 1 02. logb b 1Example 1: Evaluate each of the following logarithmsa. log4 1b. log4 4More Properties:3. If M N then logb M logb Nn4. Power Rule:logb ( M ) n logb ( M )5. Product Rule:logb ( MN ) logb M logb N M logb M logb N N 6. Quotient Rule: log b Example 2: Use log2 3 1.585 and log2 7 2.807 to evaluate each logarithm.a.log 237c. log 2 49b. log2 21d. log 2 2725
Example 3: Use the properties of logarithms to expand the following:ln5x7yExample 4: Use the properties of logarithms to expand the following: x 4 y log 7 Example 5: Use the properties of logarithms to condense the following:log 9 3log 2 log 3Example 6: Use the properties of logarithms to condense the following:13ln x ln 6 ln y426
Example 7: Use the power property to evaluate the following logarithms.a.log 5 1252b. log xx27
Name DatePre-Calculus 19-5 Properties of Logarithms – Homework1. For each of the following, use log 7 4 0.712 and log 7 12 1.277 to evaluate each logarithm.a. log 7 3b. log 7 16c. log 7 64d. log 7 1442. Draw a line to match each expression on the left with the logarithm on the right that has the same value. Justify youranswer!A. log3 6 log3 2a. log3 64B. 2log3 6b. log3 3C. 6log3 2c. log 3 12D. log3 6 log3 2d. log3 363. Use the properties of logarithms to condense the following:1log 5 4 log 5 x34. Use the properties of logarithms to condense the following:log3 2 log3 a log3 11 log3 b28
5. Use the properties of logarithms to expand the following:6. Use the properties of logarithms to expand the following:ln6 x2y4log(x3 y 4)7. Use the power property and properties of exponents to evaluate each of the following logarithms.(a. log 1023)b. ln( e)9(c. log3 811 5)8. Use the properties of logarithms to evaluate each of the following.a. ln eb. log 6 1 1 x c. log x 29
Name DatePre-Calculus 19-6 Graphing Logarithmic FunctionsTypically, we will see logarithmic functions in the formf ( x ) a logb ( x h ) k or f ( x ) a ln ( x h ) k .b is the base of the logarithma representsh representsk representsa negative in front of the x represents aThe starting point for a logarithmic function is:Asymptote: a line that a function approaches but never passes.Logarithmic functions will have an asymptote at:Example 1: Use transformations to graph each of the following functions.a.m ( x ) log3 ( x 2) 1Equation of Asymptote:Domain:Range:b. t ( x ) ln ( x 1) 3Equation of Asymptote:Domain:Range:30
c. c ( x ) log 2 ( x 3) 1d f ( x ) ln ( x 1) 4Equation of Asymptote:Equation of Asymptote:Domain:Domain:Range:Range:e.f ( x ) log2 ( x ) 3f. h ( x ) ln ( x 2 )Equation of Asymptote:Equation of Asymptote:Domain:Domain:Range:Range:31
Name DatePre-Calculus 19-6 Graphing Exponential and Logarithmic Functions – HomeworkDirections: Sketch the graph of each of the following functions. Include one accurately plotted point and label the asymptote.Then, state the domain and range.1.f ( x ) 2 x 3 3 1 4 x 12. g ( x ) 33.h ( x ) log3 ( x 4 ) 1Domain:Domain:Domain:Range:Range:Range:Equation of Asymptote:Equation of Asymptote:Equation of Asymptote: 3 4. t ( x ) 2 x 3 1 1 5. j ( x ) 2 x 4 26.z ( x ) ln ( x 3) 2Domain:Domain:Domain:Range:Range:Range:Equation of Asymptote:Equation of Asymptote:Equation of Asymptote:32
7.w ( x ) e x 48.c ( x ) 3x 4 19.z ( x ) ln ( x 4)Domain:Domain:Domain:Range:Range:Range:Equation of Asymptote:Equation of Asymptote:Equation of Asymptote:10.r ( x ) e x 1 211.p ( x ) 2x 1 412.a ( x ) log4 ( x ) 2Domain:Domain:Domain:Range:Range:Range:Equation of Asymptote:Equation of Asymptote:Equation of Asymptote:33
Name DatePre-Calculus 19-7 Solving Exponential and Logarithmic EquationsIn order to solve exponential and logarithmic equations, we need to remember how to convert between the forms. We also needto remember all of our properties of logarithms.logarithmic Form à logb y xif and only ifb x y ß exponential form32 x3Example 1: Solve the following equation:34 x 7 Example 2: Solve the following equation:8x 2 16 x 1Example 3: Solve the following equation:25 x 2Example 4: Solve the following equation:79 x 1534
Example 5: Solve the following equation:4e2 x 5 12Example 6: Solve the following equation:ln ( 4 x 7 ) ln ( x 5)Example 7: Solve the following equation:log2 (5x 17 ) 3Example 8: Solve the following equation:log2 2 log 2 (3x 5) 335
Example 9: Solve the following equation:log (3x 2) log ( 2 ) log ( x 4 )Example 10: Solve the following equation:ln x2 ln ( 2x 3)( )36
Name DatePre-Calculus 19-7 Solving Exponential and Logarithmic Equations - Homework1. Solve the following equations. 1 8 a. e2 x e3 x 1b. 5125 x 1 c. 116 x 38d. 2e 2 x 7 5 4 x2e. 256 4x 536 x 2f. 366x2. The length l (in centimeters) of a scalloped hammerhead shark can be modeled by the function l 266 219e 0.05t where tis the age (in years) of the shark. How old is the shark that is 175 centimeters long?37
3. Solve the following equations.a. ln ( 2 x 4 ) ln ( x 6 )()b. log 2 (3x 4 ) log 2 5c. log3 x2 9 x 27 2d. log6e. log5f. 7 ln ( 2 x 1) 10 5( x 4) log5 ( x 1) 2(3x ) log 6 ( x 1) 34. One hundred grams of radium are stored in a container. The amount R (in grams) of radium present after t years can bemodeled by R 100e 0.00043t . After how many years will only 5 grams of radium be present?38
Name DatePre-Calculus 19-8 Modeling with Exponential and Logarithmic FunctionsRemember:Exponential Growth:y a(1 r ) xExponential Decay:y a(1 r ) xa is the initial amountr is the percent increase/decrease (expressed as a decimal)(1 r) or (1-r) is the growth factor or the decay factort 28Example 1: The amount y (in grams) of the radioactive isotope chromium-51 remaining after t days is y a ( 0.5), where ais the initial amount (in grams). What percent of the chromium-51 decays each day?Example 2: The graph represents the temperature, in degrees Fahrenheit, of tea for the first 9 minutes after it was poured intoa cup.a. Based on the graph, what was the temperature of thetea when it was first poured into the cup?20o75o120o160ob. Based on the graph, as the number of minuteswhat temperature did the tea approach?20o75o120oincreased,160o39
Example 3: A scientist places 8.45 grams of radioactive element in a dish. The half-life of the element is 3 days. After d days, the 1 2 number of grams of the element remaining in the dish is given by the function R ( d ) 8.45 d 3. Which statement is trueabout the equation when it is rewritten without a fractional exponent? Select ALL that apply.da. An approximately equivalent equation is R ( d ) 8.45 ( 0.794)db. An approximately equivalent equation is R ( d ) 8.45 ( 0.125)c. The base of the exponent in this form of the equation can be interpreted to mean that the element decays0.794 grams per day.byd. The base of the exponent in this form of the equation can be interpreted to mean that the element decaysgrams per day.by 0.125e. The base of the exponent in this form of the equation can be interpreted to mean that about 79.4% ofelement remains from one day to the next.thef. The base of the exponent in this form of the equation can be interpreted to mean that about 12.5% ofelement remains from one day to the next.theExample 4: An investor deposits g dollars into an account at the beginning of each year for n years. The account earns anannual interest rate of r, expressed as a decimal. The amount of money S, in dollars, in the account can be determined by theformula S g n(1 r ) 1 . ra. Suppose the investor deposits 400 a year for 12 years into an account that earns an annual interest rate of 3%. Ifno additional deposits or withdrawals are made, what will be the balance in the account at the end of 12 years?b. Suppose the investor wanted the balance in the account to be at least 10,000 at the end of 12 years.annual interest rate of 3%, the amount of the initial deposit should be at least .At an40
Name DatePre-Calculus 19-8 Modeling with Exponential and Logarithmic Functions - Homework 1 1. The amount y (in grams) of the radioactive isotope Radium-226 remaining after t days is y 10 2 t 1620, where 10 is theinitial amount in grams.a. What percent of Radium-226 remains after each day?b. What percent of Radium-226 decays each day?c. How much remains after 1000 days?t2. The population P (in thousands) of Austin, Texas, during a recent decade can be approximated by P 494.29 (1.03) ,where t is the number of years since the beginning of the decade.a. Tell whether the model represents exponential growth or decay.b. Identify the annual percent increase or decrease in population.c. Estimate when the population was about 590,000.3. In 2006, there were approximately 233 million cell phone subscribers in the United States. During the next 4 years, thenumber of cell phone subscribers increased by about 6% each year.a. Write an exponential model giving the number of cell phone subscribers y (in millions) t years after 2006.Estimate the number of cell phone subscribers in 2008.b. Estimate the year when the number of cell phone subscribers was about 278 million.41
t 164. The number y of duckweed fronds in a pond after t days is y a (1230.25), where a is the initial number of fronds. Bywhat percent does the duckweed increase each day?5. The population of bacteria in a sample during a specific treatment can be modeled by the given graph, where x is the time inhours and y is the number of bacteria present.a. How many bacteria are present in thesample at the beginning of the treatment?b. As the time increases, what does thenumber of bacteria in the sample approach?c. Is this graph exponential growth or decay?6. Biologists have found that the length l in inches of analligator and its weight w in pounds are related by the function l 27.1ln ( w) 32.8.a. What is the length of an alligator that weighs 200 pounds?b. If an alligator is 10 feet long, how much does it weigh?7. A 2008 Honda sells for 17,500. A 2008 Volkswagen costs 25,000. The Honda loses 8.5% of its value each year, while theVolkswagen’s value depreciates at a rate of 10% per year. Complete the table below and answer the questions that follow.42
a. Which car is worth more after 20 years?b. After how many years does the value of the Honda cut in half?c. After how many years does the value of the Volkswagen cut in half?43
9-2 Exponential Functions Exponential Function: For any real number x, an exponential function is a function in the form fx ab( ) x. There are two types of exponential functions: Exponential Growth: fx ab b( ) x, where 1 Exponential Decay: fx ab b( ) , where 0 1
Unit 6 Exponential and Logarithmic Functions Lesson 1: Graphing Exponential Growth/Decay Function Lesson Goals: Identify transformations of exponential functions Identify the domain/range and key features of exponential functions Why do I need to Learn This? Many real life applications involve exponential functions.
The Natural Logarithmic and Exponential The Natural Logarithmic and Exponential and Exponential Function FunctionFunctions sss: . Differentiate and integrate exponential functions that have bases other than e. Use exponential functions to model compound interest and exponential
348 Chapter 7 Exponential and Logarithmic Functions 7.1 Lesson WWhat You Will Learnhat You Will Learn Graph exponential growth and decay functions. Use exponential models to solve real-life problems. Exponential Growth and Decay Functions An exponential function has the form y abx, where
Chapter 3 Exponential & Logarithmic Functions Section 3.1 Exponential Functions & Their Graphs Definition of Exponential Function: The exponential function f with base a is denoted f(x) a x where a 0, a 1 and x is any real number. Example 1: Evaluating Exponential Expressions Use
Section II: Exponential and Logarithmic Functions Unit 1: Introduction to Exponential Functions Exponential functions are functions in which the variable appears in the exponent. For example, fx( ) 80 (0.35) x is an exponential function si
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218 Chapter 3 Exponential and Logarithmic Functions What you should learn Recognize and evaluate expo-nential functions with base a. Graph exponential functions and use the One-to-One Property. Recognize, evaluate, and graph exponential functions with base e. Use exponential
Choir Director: Ms. Cristy Doria Organist: Dr. Devon Howard Choir Accompanists: Madison Tifft & Monte Wilkins After the benediction, please be seated as the graduates leave the sanctuary. The classes of 2018 & 2019 are hosting an invitation-only dinner in the Fellowship Hall in honor of the graduates and their families. Special Thanks to
