Hartfield College Algebra (Version 2018 - Thomas Hartfield) Unit FIVE Page - 1 - of 39 Topic 37: Exponential Functions In previous topics we ve discussed power functions, n functions of the form f x x, where n is a nonzero real number. Squaring and cubing functions, along with all of the even or odd powered functions, are obviously power functions, but so are root functions (n is a rational number for the power) and reciprocal functions (n is a negative integer). Another enormous super-family of functions are exponential functions. Definition: As exponential function with base b f x b x b b is defined as, 0, 1. Domain:, Range: 0,
Hartfield College Algebra (Version 2018 - Thomas Hartfield) Unit FIVE Page - 2 - of 39 Examples of Basic Exponential Functions Observations about exponential functions: 1. f(0) = 1. All basic exponential functions have a y-intercept at (0, 1). 2. All basic exponential functions have a horizontal asymptote of y = 0. f x 2 x f x 5 x 3a. When b > 1, a basic exponential function is always increasing. The right end goes up (as x the function increases without bound) and the left end asymptotically approaches the x-axis (as x the function approaches 0). 3b. When 0 < b < 1, a basic exponential function is always decreasing. The left end goes up (as x the function increases without bound) and the right end asymptotically approaches the x-axis (as x the function approaches 0). 4. The closer b is to 1, the more linear the graph appears. f x 0.5 x f x 0.8 x
Hartfield College Algebra (Version 2018 - Thomas Hartfield) Unit FIVE Page - 3 - of 39 Natural Exponential Functions and the Number e When discussing exponential functions, it is important to introduce a special number that shows up frequently in mathematics, the number e. Definition: n 1 e lim 1 2.71828182846 n n A very special basic exponential function is one which uses e as its base. Definition: The natural exponential function is an exponential function with base e: x e. f x Now what does this definition mean? A limit (the abbreviation lim) describes the behavior of an expression as a variable (in this case n) changes in a certain way. If n gets larger and larger, without ceasing, the expression here gets closer and closer to the value of e. The function behaves all of the basic rules of exponential functions, including the fact that the domain is, and the range is 0,, and since e > 1, it satisfies observation 3a made previously. We refer to e by several names, including Napier s Constant or Euler s Number, but most frequently it is identified as the Natural Base.
Hartfield College Algebra (Version 2018 - Thomas Hartfield) Unit FIVE Page - 4 - of 39 Transformations of basic exponential functions The algebraic transformations we covered in a previous unit are possible with exponential functions as well. Ex. 1 Identify the basic function in the given function. Determine the transformations applied to the basic function that produce the given function. State the transformations applied, in the correct order, using units and directions as appropriate. Finally, determine the domain, range, and asymptote of the given function. f( x) 32 x 2 Ex. 2 Identify the basic function in the given function. Determine the transformations applied to the basic function that produce the given function. State the transformations applied, in the correct order, using units and directions as appropriate. Finally, determine the domain, range, and asymptote of the given function. f( x) 63 x 1
Hartfield College Algebra (Version 2018 - Thomas Hartfield) Unit FIVE Page - 5 - of 39 Ex. 3 Determine the simplest exponential function that satisfies the conditions below. Ex. 4 Determine the simplest exponential function that satisfies the conditions below. a. Has a y-intercept of (0, 1) and passes 1 through the point 1,. 2 a. Has a y-intercept of (0, 4) and passes b. Has a y-intercept of (0, 1) and passes 9 through the point 2,. 4 b. Has a y-intercept of (0, 3) and passes
Hartfield College Algebra (Version 2018 - Thomas Hartfield) Unit FIVE Page - 6 - of 39 Topic 38: Logarithmic Functions & Expressions Since basic exponential functions are one-to-one, that means they are invertible. This leads us to the definition of another super-family of functions which are the inverses of exponentials. Ex. 1 Using the graph of f( x) 2 x below, sketch the graph of gx ( ) log 2 x. Definition: The inverse of an exponential x function f x b is called a base b logarithmic function and notated as f 1 x log b x (which is read as log base b of x).
Hartfield College Algebra (Version 2018 - Thomas Hartfield) Unit FIVE Page - 7 - of 39 below, 2 sketch the graph of gx ( ) log 1 x. 2 1 Ex. 2 Using the graph of f( x) x Conclusions that can be drawn about logarithmic functions. 1. Domain is 0, and Range is,. 2. g(1) = 0. All basic logarithmic functions have an x-intercept at (1, 0) and have a vertical asymptote of x = 0. 3. When b > 1, a basic logarithmic function is always increasing and when 0 < b < 1, a basic logarithmic function is always decreasing. Two special types of logarithmic functions are the common logarithm whose base is 10, log 10 x, which is usually written as just log x and the natural logarithm whose base is e, log e x, which is usually written as just ln x.
Hartfield College Algebra (Version 2018 - Thomas Hartfield) Unit FIVE Page - 8 - of 39 Transformations are also possible with logarithmic functions. Ex. 3 Identify the basic function in the given function. Determine the transformations applied to the basic function that produce the given function. State the transformations applied, in the correct order, using units and directions as appropriate. Finally, determine the domain, range, and asymptote of the given function. f( x) log x 3 2 Ex. 4 Identify the basic function in the given function. Determine the transformations applied to the basic function that produce the given function. State the transformations applied, in the correct order, using units and directions as appropriate. Finally, determine the domain, range, and asymptote of the given function. f( x) 5log x 1 2
Hartfield College Algebra (Version 2018 - Thomas Hartfield) Unit FIVE Page - 9 - of 39 Let s explore logarithmic expressions (which we can just call logarithms) in depth. An essential property of logarithms is the following: n b c log b( c) n. That is, logb (c) is the exponent you apply to b to make c. Ex. 6 Evaluate each logarithm. a. log6 36 Ex. 5 Evaluate each logarithm. a. log7 1 b. log3 81 b. log1111 c. log25 5
Hartfield College Algebra (Version 2018 - Thomas Hartfield) Unit FIVE Page - 10 - of 39 Ex. 7 Evaluate each logarithm. Ex. 8 Evaluate each logarithm. a. log9 27 a. log32 4 1 b. log 100 1 b. log 12 12 c. log0.5 16 c. log5 0.04
Hartfield College Algebra (Version 2018 - Thomas Hartfield) Unit FIVE Page - 11 - of 39 Topic 39: Properties of Logarithms In addition to the property of logarithms established previously, there are three additional properties of logarithms (similar to properties of exponentials and sometimes called laws of logarithms ): Product Rule for exponentials: m n m n a a a Product Rule for logarithms: log xy log x log y a a a It is possible to take a single relatively complicated logarithm and rewrite it into a collection of simpler logarithmic expressions or constants by using the properties of logarithms. Ex. 1 Expand the expression using the properties of logarithms. Simplify where possible. log2 4 8x y a mn Quotient Rule for exponentials: a n a Quotient Rule for logarithms: log x log x log y m Power Rule for exponentials: n n Power Rule for logarithms: loga m a a a a m n a x nlog x a y Deconstructing a single logarithm:
Hartfield College Algebra (Version 2018 - Thomas Hartfield) Unit FIVE Page - 12 - of 39 Ex. 2 Expand the expression using the properties of logarithms. Simplify where possible. Ex. 3 Expand the expression using the properties of logarithms. Simplify where possible. 2 ln a ec 5 log 7 x 5 6 2 y z
Hartfield College Algebra (Version 2018 - Thomas Hartfield) Unit FIVE Page - 13 - of 39 Ex. 4 Expand the expression using the properties of logarithms. Simplify where possible. 9 v log3 9 n Reconstructing a single logarithm: This is the opposite of the last skill. Sometimes a single logarithm may be a preferable expression. Ex. 5 Rewrite as a single logarithm. 3log x4log y 5log z 11 11 11 Ex. 6 Rewrite as a single logarithm. log 2 6log ulog 7 7 7 v
Hartfield College Algebra (Version 2018 - Thomas Hartfield) Unit FIVE Page - 14 - of 39 Ex. 7 Rewrite as a single logarithm. Ex. 9 Rewrite as a single logarithm. 3log x log x 2 log x 1 log x 1 2 2 2 2 log xlog x 2 3 1 3 Ex. 8 Rewrite as a single logarithm. 2log x log x1 log 3 log 2x 1 1 4 2 4 4 4 Ex. 10 Rewrite as a single logarithm. log xlog x 2 3 9
Hartfield College Algebra (Version 2018 - Thomas Hartfield) Unit FIVE Page - 15 - of 39 Using the properties of logarithms it should be relatively easy to evaluate log2 32. Change of Base Formula log a n log log b b n a Calculators will easy handle approximating log32 and ln32. But how do we determine the approximate value of log 32? 3 At the very least we should be able to determine between which two integers the value should fall. This formula allows us to calculate the approximate value of logarithms where the exact value is difficult or impossible to express as a non-logarithmic expression. Since we can choose what we change the base into, the most logical choices are usually base 10 or base e, thus: log a n ln ln n a log log n a Ex. 11 Approximate the value of 3 log 32, five places after the decimal point.
Hartfield College Algebra (Version 2018 - Thomas Hartfield) Unit FIVE Page - 16 - of 39 Topic 40: Solving Logarithmic Equations Ex. 1 Solve. A logarithmic equation is an equation involving one or more logarithmic expressions. x x log 5 1 log 1 1 8 8 Logarithmic equations can be divided into two families: 1. Those equations that can be simplified into the form of a logarithmic expression equal to a number. 2. Those equations that can be simplified into the form of two logarithmic equations (same base) equal to each other. To solve the first case, apply the following steps: 1. Use the laws of logarithms, as necessary, to combine logarithms. 2. Rewrite the logarithm equation into exponential form. 3. Solve using any appropriate algebraic and/or arithmetic methods. 4. Check your solutions. Like radical equations, logarithmic equations can (and often do) produce extraneous solutions.
Hartfield College Algebra (Version 2018 - Thomas Hartfield) Unit FIVE Page - 17 - of 39 Ex. 2 Solve. Ex. 3 Solve. x x log 7 log 5 3 4 4 log 3 x 1 5
Hartfield College Algebra (Version 2018 - Thomas Hartfield) Unit FIVE Page - 18 - of 39 Ex. 4 Solve. x x log 1 log 3 5 2 2 If the equation is simplified (using the laws of logarithms) in such a way both sides have a logarithmic expression, consider the limited possibilities for making the equation true... Ex. 5 Solve. x x ln 2 ln 3 ln 4
Hartfield College Algebra (Version 2018 - Thomas Hartfield) Unit FIVE Page - 19 - of 39 Ex. 6 Solve. x x x x log 4 log 5 log 1 log 7
Hartfield College Algebra (Version 2018 - Thomas Hartfield) Unit FIVE Page - 20 - of 39 Ex. 7 Solve. x x x x log 1 log 2 log log 7 3 3 3 3
Hartfield College Algebra (Version 2018 - Thomas Hartfield) Unit FIVE Page - 21 - of 39 Topic 41: Solving Exponential Equations (Part 1) Ex. 1 Solve. An exponential equation is an equation involving one or more exponential expressions. 3x 6 100 Basic process to solve most exponential equations: 1. Isolate an exponential expression on one side. 2. Take the natural logarithm (or common log) of both sides. 3. Use the laws of logarithms to rewrite the exponential expression so that no variable remains in the exponent. 4. Apply basic algebraic and arithmetic manipulation to solve for x. 5. Use the laws of logarithms to simplify the solution and approximate the solution. 6. Check your solution. With regards to step 5, I will expect you to further rewrite the solution into a single logarithmic expression and approximate five places after the decimal.
Hartfield College Algebra (Version 2018 - Thomas Hartfield) Unit FIVE Page - 22 - of 39 Ex. 2 Solve. Ex. 3 Solve. x2 43 16 2x 1 5 18
Hartfield College Algebra (Version 2018 - Thomas Hartfield) Unit FIVE Page - 23 - of 39 Ex. 4 Solve. Ex. 5 Solve. 2x 1 10 500 x 5e 3 20
Hartfield College Algebra (Version 2018 - Thomas Hartfield) Unit FIVE Page - 24 - of 39 Topic 42: Solving Exponential Equations (Part 2) Ex. 2 Solve. If the bases are equal or the bases are both powers of the same constant, solving an equation of two exponential expressions is fairly easy. x2 2x1 8 4 Ex. 1 Solve. x3 2x1 5 5
Hartfield College Algebra (Version 2018 - Thomas Hartfield) Unit FIVE Page - 25 - of 39 If the bases are neither equal nor related, the process for solving an exponential equation involving two exponential expressions is similar (but more involved) than those covered in a previous topic. Ex. 3 Solve. 2x3 x1 3 2
Hartfield College Algebra (Version 2018 - Thomas Hartfield) Unit FIVE Page - 26 - of 39 Ex. 4 Solve. x1 2x 4 3
Hartfield College Algebra (Version 2018 - Thomas Hartfield) Unit FIVE Page - 27 - of 39 Some exponential equations be solved using factoring (with substitution sometimes helpful). The important detail to remember here is 2x x that 2 e e. Ex. 6 Solve. e 2x x 4e 0 Ex. 5 Solve. e 2x x e 20
Hartfield College Algebra (Version 2018 - Thomas Hartfield) Unit FIVE Page - 28 - of 39 Topic 43: Compound Interest There are two primary formulas used for problems involving compounded interest. Formula 1 Discretely Compounded Interest P1 At r n nt t = units of time in years P = principle (initial value) r = rate of interest as a decimal A(t) = amount at/after time t n = number of compounds per year This is the formula used most often when interest is compounded. In this formula compounding occurs at some defined regular intervals. Examples include quarterly (n = 4), monthly (n = 12), and weekly (n = 52), amongst others. A second formula for compounding interest, which involves the natural base, is used only in cases where interest is compounded continuously. Formula 2 Continuously Compounded Interest At r t Pe t = units of time in years P = principle (initial value) r = rate of interest as a decimal A(t) = amount at/after time t
Hartfield College Algebra (Version 2018 - Thomas Hartfield) Unit FIVE Page - 29 - of 39 Given information about all constants in the formulas except one, it is possible to solve for that unknown. For example: If given everything but A(t), you can find a future amount from a present amount. If given everything but P, you can find the present amount needed for a future amount. If given everything but t, you can solve an equation to find the time needed to go from a present amount to a future amount. Ex. 1a Jordan wishes to invest $2 500. He is looking at two different Certificates of Deposit (CDs), each of which compounds interest monthly. The two year CD earns 1.20% interest while the five year CD earns 2.25% interest. How much will each CD be worth at maturation? Express to the penny.
Hartfield College Algebra (Version 2018 - Thomas Hartfield) Unit FIVE Page - 30 - of 39 Ex. 1b If Jordan wants to have $3000 in the future with the five year CD, how much would he have to invest now? Express to the penny. Ex. 1c If Jordan is offered a savings account earning 1.5% interest compounded quarterly, how long would it take a $2500 investment to double? Express to the nearest tenth of a year.
Hartfield College Algebra (Version 2018 - Thomas Hartfield) Unit FIVE Page - 31 - of 39 Ex. 2a A furniture company offers a promotion where no interest is accrued on an account for four years. In the fine print you find that if the account is not paid in full within the four year period, all of the interest that could have accumulated on the original amount will retroactively be applied to the account. A $3 000 sofa is not paid off in time. How much interest would be tacked on to the account if the interest is 15.99% compounded biweekly? Express to the dollar. A term often used in finance, associated with lending and savings, is Annual Percentage Rate, or APR. The APR of an account is a measure of what an equivalent rate would be if the interest was only compounded annually (this provided a clearer picture on the long term effect of the nominal rate.) To calculate the APR on an account, let P = $1 and t = 1 year. Then subtract the original dollar from A(1) and convert to a percentage. Ex. 2b What is the APR on the furniture company s promotional offer? Express to the nearest hundredth of a percent.
Hartfield College Algebra (Version 2018 - Thomas Hartfield) Unit FIVE Page - 32 - of 39 Ex. 3a Ashley has invested $4000 in an interest-bearing savings account which earns 3.75% interest compounded quarterly. How much will she have in 5 years? Express to the penny. Ex. 3b How long will it take for Ashley s investment to be worth $10,000? Express to the tenth of a year.
Hartfield College Algebra (Version 2018 - Thomas Hartfield) Unit FIVE Page - 33 - of 39 Ex. 4 Taylor s grandmother put away money for him in a municipal bond on the day he was born, with instructions that he should only cash it in when it had tripled in value. If the bond earns 5.2% interest, compounded weekly, how old will he be when he can cash it in according to her wishes? Ex. 5 Shannon wants her investment of $10,000 to be valued at $30,000 in 10 years. What interest rate, expressed to the hundredth of a percent, will accomplish this if the interest is compounded continuously?
Hartfield College Algebra (Version 2018 - Thomas Hartfield) Unit FIVE Page - 34 - of 39 Topic 44: Other Applications of Exponentials Radioactive Decay A variety of application problems can use formulas with exponentials in them. Population Growth based on a continuous rate The growth (or decline of a population) can be r t determined by the formula n t n 0 e. n 0 = initial population n(t) = population at time t t =time (any units) r = relative rate of growth per unit of time The amount of radioactive material in a sample can be determined by the formula 0 2 mt m t h. m 0 = initial amount of radioactive material m(t) = amount of radioactive material at time t t = time h = half-life of radioactive material (units for time and half-life must agree) Newton s Law of Cooling The temperature of an item after being in an environment of a different temperature can be kt found by the formula T t AT Ae 0. T 0 = initial item temp. T(t) = temperature at time t A = temperature of the environment t = time (any units) k = cooling rate
Hartfield College Algebra (Version 2018 - Thomas Hartfield) Unit FIVE Page - 35 - of 39 Ex. 1 An abandoned island, overrun with rats, is used as a dumping ground. The rat population is estimated by the function 0.15t n t 45 e, where t is the number of years after 2005 & n(t) is in thousands. a: According to the formula, what was the population of rats in 2005? Ex. 2 According to projections by the state of Georgia, Forsyth County had about population of 183 thousand in 2010 and is expected to have a population of about 264 thousand in 2020. What is the expected relative rate of growth in Forsyth County over the decade from 2010 to 2020? Express to the tenth of a percent. b: According to the formula, what was the relative growth rate in the rat population? c: Estimate the number of rats on the island in 2025. Express to the tenth of a thousand.
Hartfield College Algebra (Version 2018 - Thomas Hartfield) Unit FIVE Page - 36 - of 39 Ex. 3 Morgan wishes to do an experiment with fruit flies. At 9 am he has 200 flies and under the controls of his experiment their population should increase at a relative rate of 35% per hour. a: Estimate the population of fruit flies at 4 pm of the same day. Express to the whole fly. b: At what time, to the nearest minute, will his population reach 5000 flies?
Hartfield College Algebra (Version 2018 - Thomas Hartfield) Unit FIVE Page - 37 - of 39 Ex. 4 Modern smoke detectors use gamma rays produced by Americium-241 to help detect smoke at earlier stages. The half-life of Am-241 is 432 years. If 0.5 grams of Am-241 is used, how much will remain after 10 years when the life of the smoke detector is exhausted (due to mechanical and environmental wear)? Approximate to three decimal places. Ex. 5 Technetium-99m is used in many medical applications including as a tracer for diagnosing bone infections in children. The half-life of Te-99m is 6.02 hours. Riley took a dose of Te-99m at 6am this morning. How much of it remains in her system at 9pm tonight? Express to the tenth of a percent.
Hartfield College Algebra (Version 2018 - Thomas Hartfield) Unit FIVE Page - 38 - of 39 Ex. 6 A new treatment for cancer that targets bone metastasis involves the use of Radium-223. The half-life of Ra-223 is 11.43 days. Leslie can only receive new treatments when 90% of the previous application has exited his body. How many days must Leslie wait between treatments? Ex. 7 A cup of coffee brewed at 200ºF is served to a customer at a football game where the temperature is 50ºF. Based on the material of the cup, the cooling rate for the cup is 2 percent per minute. Find the model for the temperature of the coffee, where time is in minutes, and then determine the temperature of the coffee an hour later.
Hartfield College Algebra (Version 2018 - Thomas Hartfield) Unit FIVE Page - 39 - of 39 Ex. 8 Robin heats water to 190ºF in a kitchen whose temperature is 70ºF. In 6 minutes the temperature of the water is 170ºF. a: Find the formula for the temperature of the water t minutes after it is removed from the stove top. b: In minutes and seconds hsow long will it take the water to cool to 100ºF from the original 190ºF?