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Math Analysis Chapter 3 Notes: Exponential and Logarithmic Functions Day 21: Section 3-1 Exponential Functions 3-1: Exponential Functions After completing section 3-1 you should be able to do the following: 1. Evaluate exponential functions 2. Graph exponential functions 3. Evaluate functions with base e 4. Use compound interest formulas An exponential functions are functions whose equations contain a variable in the exponent. Definition of the Exponential Function The exponential function f with base b is defined by: f ( x) b x or y b x Where b is a positive constant other than 1 (b > 0 and b ≠ 0) and x is any real number. Example of exponential functions: f ( x) 3 x base of 3 h( x) 10 e( x) 2 x base of 10 x 1 1 k ( x) 2 x 3 base of 2 base of 1/2 Example of functions that are not exponential functions: a( x) 1x The base of an exponential function must be a positive constant other than 1. b ( x ) 2 x The base of an exponential function must be a positive. c( x) x4 Variable is the base and not the exponent. d ( x) x x Variable is both the base and the exponent. To evaluate expressions with exponents with a calculator: 1. Enter base 2. find the button [^] or [yx] and push it 3. enter exponent 4. push the equal button and you should have your answer. Practice: Approximate each number using a calculator. Round your answer to three decimal places. 1. 32.5 2. 4 6 3. e 4 4. 63.1 1 Math Analysis Notes Chapter 3 Mr. Hayden Graphing Exponential Functions Graphing exponential functions in the form y = ab x for b > 1 where a is a real number and b is the base (b ≠ 1) Practice: (a) Graph each exponential function. (b) State the domain and range in interval notation. (c) Label the horizontal asymptote. 1. y = 4x x y 2. y = x 1 x 4 2 y Graphing Exponential Functions in the form y = abx − h + k You must graph the parent function 1st. y = abx Then translate the graph horizontally according to h and vertically according to k. You must show both the parent function and the translated function in order to get credit when graphing exponential functions that have translations in them. Practice: (a) Graph each exponential function. (b) State the domain and range in interval notation. (c) Label the horizontal asymptote. 2. y = 2 x 3 1 1. y =4•2x − 1 – 3 x y x y 2 Math Analysis Notes Chapter 3 Mr. Hayden The Natural Base e An irrational number, symbolized by the letter e, appears as the base in many applied exponential functions. The number e is n 1 defined as the value that 1 as n gets larger and larger. As n goes to ∞ the approximate value of e to nine decimal places n is: e ≈ 2.718281827. The irrational number e approximately 2.72, is called the natural base. The function x f ( x ) e is called the natural exponential function. Practice: (a) Graph each exponential function. (b) State the domain and range in interval notation. (c) Label the horizontal asymptote. 2. y = e x 3 1. y = ex x y x y Formulas for Compound Interest After t years, the balance A, in an account with principal P and annual interest rate r (in decimal form) is given by the following formulas: 1. r For n compoundings per year: A P 1 n 2. For continuous compounding: A Pert nt Practice: A sum of $10,000 is invested at an annual rate of 8%. Find the balance in the account after 5 years subject to (a) quarterly compounding and (b) continuous compounding. 3 Math Analysis Notes Chapter 3 Mr. Hayden Geometry Review: Trigonometry A ratio of the lengths of two sides of a right triangle is called a trigonometric ratio. The three most common ratios are sine, cosine, and tangent. These three rations are defined for the acute angles of right triangles, though your calculator will give you values of sine, cosine, and tangent for angles of greater measure. The abbreviations for the ratios are sin, cos, and tan respectively. sin A leg opposite to A opp a hypotenuse hyp c cos A leg adjacent to A adj b hypotenuse hyp c tan A leg opposite to A opp a leg adjacent to A adj b B c a A b C In 1-6, find the indicated trigonometric ratio using the right triangles to the right. Final answers should be in reduced fractional form. 1. sinM 2. cosZ Y K 3. tanL 4 4. sinX 6 3 6 5. cosL 6. tanZ Z X 4 2 L M In 7-10, find the trigonometric ratio that corresponds to each value and the angle given, using the triangle at the right. 7. 9 , G 41 8. 40 , H 9 H 41 9 9. 40 , H 41 10. 9 , H 41 E 40 G 4 Math Analysis Notes Chapter 3 Mr. Hayden Day 23: Section 3-2 Logarithmic Functions 3-2: Logarithmic Functions After completing section 3-2 you should be able to do the following: 1. Change from logarithmic to exponential form. 2. Change from exponential to logarithmic form. 3. Evaluate logarithms. 4. Use basic logarithmic properties. 5. Graph logarithmic functions 6. Find the domain and range of a logarithmic function. 7. Use of common logarithms 8. Use natural logarithms Definition of the Logarithmic Function For x > 0 and b > 0, b ≠ 1: y log b x is equivalent to b y x The function f ( x) logb x is the logarithmic function with base b. The point of logarithmic functions is they allow us to solve for the value of a variable that is an exponent. Exponent Exponent Exponential Form: by = x Logarithmic Form: y = logbx Base Base To change from logarithmic form to the more familiar exponential form, use this pattern: y = logbx means by = x Practice: In 1-4, Write each equation in its equivalent exponential form: 1. 3 = log7x 2. 2 = logb25 3. log426 = y 4. log28 = x Practice: In 1-4, Write each equation in its equivalent logarithmic form: 1. 25 = 32 2. b3 = 27 3. ey = 33 4. 4x = 64 To Evaluate a logarithmic expression without using a calculator: 1. Set logarithmic expression equal to x 2. Write the equation in its equivalent exponential form 3. Evaluate the exponential expression 4. The answer to the exponential expression it the value of the logarithmic expression. Practice: In 1-4, Evaluate each expression without using a calculator. 1. log416 2. log648 3. log264 1 4. log 4 16 5 Math Analysis Notes Chapter 3 Mr. Hayden Basic Logarithmic Properties Involving One 1. logb b 1 because 1 is the exponent to which b must be raised to obtain b. logb 1 0 because 0 is the exponent to which b must be raised to obtain 1. (b0 = 1) 2. Inverse Properties of Logarithms For b > 0 and b ≠ 1: log b b x x The logarithm with base b of b raised to a power equal that power blogb x x b raised to the logarithm with base b of a number equals that number Practice: In 1-4 Evaluate each expression without using a calculator. 1. log99 2. log41 3. log773 4. 8log819 To Graph a logarithmic Functions in the form y = log bx 1. Rewrite the function in exponential form 2. Graph the exponential equation by making an x/y table. You will be choosing values for the exponent (in this case y) 3. Connect points with a smooth curve. Practice: Graph y = 2x and y = log2x on the same rectangular coordinate system. y = 2x x y = log2x y x y Characteristics of graphs of Logarithmic Functions Have vertical asymptotes Domain is restricted by vertical asymptote, however, range is , 6 Math Analysis Notes Chapter 3 Mr. Hayden To Graph a logarithmic Functions in the form y = alog b(x – h) + k 1. Write logarithmic function that does not contain transformations h or k. 2. Write the logarithmic function found in step 1 in exponential form. 3. Follow steps above to graph the exponential function found in step 2. 4. Now use h to translate each point horizontally h-units and k to translate each point vertically k-units 5. Connect these new points with a smooth curve to get the graph of y = alog b(x – h) + k Practice: (a) graph: y = 3 – 2log3(x – 1). (b) State the domain and range. (c) Write the equation of the asymptote of the graph. Practice: (a) graph: y = ln(x + 1) – 3. (b) State the domain and range. (c) Write the equation of the asymptote of the graph. The common base (10): The natural base (e): A logarithm with a base of 10 is written without a base. So log15 is read as “log base 10 of 15” or “the common log of 15” A logarithm with a base of e is written a natural logarithm. So loge15 is written as ln15 and is read as “natural log of 15” 7 Math Analysis Notes Chapter 3 Mr. Hayden Practice: Evaluate or simplify each expression without using a calculator. 1. log100 2. lne 3. lne8 4. log104x Geometry Review: Trigonometry Besides the three most common trigonometric ratios, sine, cosine, and tangent, there are three more rations that are considered the reciprocal ratios. These reciprocal ratios are cosecant, secant, and cotangent. The abbreviations for the ratios are csc, sec, and cot respectively. csc A hypotenuse hyp c 1 leg opposite to A opp a sin A sec A hypotenuse hyp c 1 leg adjacent to A adj b cos A cot A leg adjacent to A adj b 1 leg opposite to A opp a tan A B c a A b C In 1-6, find the indicated trigonometric ratio using the right triangles to the right. Final answers should be in reduced fractional form. 1. cscM 2. secZ Y K 3. cotL 4 4. cscX 6 5. secL 6. tanZ 6 3 Z X 4 2 L M How to find another trigonometric equation given one trigonometric equation. Use the given information to draw a right triangle and making the given sides Find the missing side using Pythagorean Theorem Now that you have all there sides of the right triangle labeled you can write the trigonometric equation for any ratio. In 7-: Use the given trig equation to find the value of a different trig ratio. 4 5 7.) sin x , cos x ? 8.) tan x , secx ? 5 12 1 5 9.) cos x , cscx ? 10.) sin x , cot x ? 2 7 8 Math Analysis Notes Chapter 3 Mr. Hayden Day 24: Section 3-3 Properties of Logarithms; Section 3-4 Exponential and Logarithmic Equations 3-3: Properties of Logarithms After completing section 3-3 you should be able to do the following: 1. Use the product rule 2. Use the quotient rule 3. Use the power rule 4. Expand logarithmic expressions 5. Condense logarithmic expressions 6. Use the change-or-base property Rules of Logarithms (very similar to the rules of exponents) Let b, M, and N be positive real numbers with b ≠ 1. The Product Rule logb(MN) = logbM + logbN The logarithm of a product is the sum of the logarithms. The Quotient Rule M logg log b M log b N N The logarithm of a quotient is the difference of the logarithms. The Power Rule log b M x xlog b M The logarithm of a number with an exponent is the product of the exponent and the logarithm of that number. We use these rules in order to expand or condense a logarithmic expression Practice: 1-4, use the properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate the logarithmic expression without using a calculator. 8x2 3 1 x x 1. log7(7x) 2. log 3. log 100x 4. log 2 2 7 x 1 100 Practice: 1-3, use the properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. 1. log(2x – 5) – 3log3 2. 3(logx + logy) – 2(log(x + 1)) 3. 4lnx + 7lny – 3lnz 9 Math Analysis Notes Chapter 3 Mr. Hayden Using Change of base to evaluate Logarithms Calculators can only evaluate logarithms that have the common base (10) or the natural base (e). We can change any base of a logarithm by using the change of base property: Changing to the Common Base logM log b M logb Changing to the Natural Base lnM log b M lnb Practice: Use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. 1. log513 2. log1487.5 3. log0.112 4. logπ60 3-4: Exponential and Logarithmic Equations After completing section 3-4 you should be able to do the following: 1. Use like bases to solve exponential equations 2. Use logarithms to solve exponential equations 3. Use the definition of a logarithm to solve logarithmic equations 4. Use the one-to-one property of logarithms to solve logarithmic equations. Two methods to solving exponential equations: Method 1: Expressing each side as a power of the same base. Practice: In 1-4, Solve: 1. 53x – 6 = 125 2. 8x + 2 = 4x – 3 3. 5x = 1 25 4. 6 x 3 4 6 Method 2: Using Natural Logarithms to Solve Exponential Equations Since most exponential equations cannot be rewritten so that each side has the same base. Logarithms are extremely useful in solving such equations. Steps to solve exponential equations using natural logarithms 1. Isolate the exponential expression. 2. Take the natural logarithm on both sides of the equation. 3. Simplify using one of the following properties: lnbx = xlnb or lnex = x 4. Solve for the variable. Practice: In 1-4 Solve: 10 Math Analysis Notes Chapter 3 Mr. Hayden 1. 5x = 134 2. 7e2x – 5 = 58 3. 32x – 1 = 7x + 1 4. e2x – 8ex + 7 = 0 Logarithmic Equations Steps to solve logarithmic equations 1. Get logarithm on one side of the equation and make sure the coefficient is 1. If not use algebraic properties to move constants or coefficients to the other side of the equal sign if necessary. 2. Use the properties of logarithms to write the expression as a single logarithm whose coefficient is 1. (Condense if necessary) 3. Use the definition of a logarithm to rewrite the equation in exponential form: logbM = c means bc = M 4. Solve for the variable 5. Check proposed solutions in the original equation. Include in the solution set only values for which M > 0 Practice: In 1-4 Solve: 1. log2(x – 4) = 3 2. 4ln(3x) = 8 3. logx + log(x – 3) = 1 4. ln( x 3) ln(7 x 23) ln( x 1) More Trig Review A ratio of the lengths of two sides of a right triangle is called a trigonometric ratio. The six trigonometric ratios are defined for the acute angles of a right triangle as: sin A leg opposite to A opp a hypotenuse hyp c csc A hypotenuse hyp c 1 leg opposite to A opp a sin A cos A leg adjacent to A adj b hypotenuse hyp c sec A hypotenuse hyp c 1 leg adjacent to A adj b cos A tan A leg opposite to A opp a leg adjacent to A adj b cot A leg adjacent to A adj b 1 leg opposite to A opp a tan A B c a A b C 11 Math Analysis Notes Chapter 3 Mr. Hayden A harmonic that can be used to remember the 1st three trigonometric ratios: sine, cosine, and tangent is SOH-CAH-TOA. To remember the reciprocal functions cosecant, secant, and cotangent you can use “HO”, “HA” and “AO” respectively. Practice: In 1-2, Use the given trigonometric equation to find the remaining five trigonometric equations. 6 3 1.) sin X 2.) csc G 2 4 Practice: In 3-6, Use the given trigonometric equation to find the indicated trigonometric ratio. 3.) sec X 2, cot X ? 2 , tan X ? 4.) sin X 2 5.) csc X 5, cos X ? 3 6.) tan X , sin X ? 2 12 Math Analysis Notes Chapter 3 Mr. Hayden Day 25: Section 3-5 Exponential Growth and Decay; Modeling Data; Compound Interest Problems 3-5: Exponential Growth and Decay After completing section 3-5 you should be able to do the following: 1. Model exponential growth and decay 2. Use compound interest formulas to solve word problems Exponential Growth and Decay Models The mathematical model for exponential growth or decay is given by: A = a0ekt Where a0 = original amount, or size of the growing or decaying entity at t = 0. A is the amount at time t. and k is a constant representing the growth rate (many times given as a percentage). If k is positive the function models a growth If k is negative the function models a decay Exponential Growth k>0 Exponential Decay k<0 Practice: The exponential model A = 106.2e.018t describes the population of a country, A, in millions, t years after 2003. Use this model to solve Exercises 1-4. 1. What was the population of the country in 2003? 2. Is this county’s having a population growth or decay? 3. What will be the population in 2012? 4. When will the population be 1000 million? 13 Math Analysis Notes Chapter 3 Mr. Hayden Formulas for Compound Interest After t years, the balance A, in an account with principal P and annual interest rate r (in decimal form) is given by the following formulas: r For n compoundings per year: A P 1 n nt For continuous compounding: A Pert Practice: 1. Find the number of years it takes for $10,000 to double at an interest rate of 7% compounded quarterly. 2. Find the number of years it takes $1500 to become $4000 at an interest rate of 5.5% compounded continuously. 14 Math Analysis Notes Chapter 3 Mr. Hayden More Trig Review Remember to use SOH-CAH-TOA & HO-HA-A0 to find the six trigonometric ratios. Hypotenuse A Si de Op posi te to A Si de Adj acent to A sin A opp 1 ( SOH ) hyp csc A csc A hyp 1 ( HO) opp sin A cos A adj 1 (CAH ) hyp sec A sec A hyp 1 ( HA) adj cos A tan A opp 1 (TOA) adj cot A cot A adj 1 ( AO) opp tan A Practice: In 1-2, Use the given trigonometric equation to find the remaining five trigonometric equations. 2 5 1.) cos X 2.) cot G 6 3 Practice: In 3-6, Use the given trigonometric equation to find the indicated trigonometric ratio. 3.) csc X 3, tan X ? 1 4.) sin K , tan K ? 2 5 5.) sec X , sin X ? 2 3 6.) cot X , sec X ? 2 15 Math Analysis Notes Chapter 3 Mr. Hayden Chapter 3 Review Sheet Please complete each of the following problems on a separate sheet of paper. Show all of your work! NO WORK = NO CREDIT! For questions 1-4, graph each function by making a table of values. State the domain, range, and equations of any asymptotes. 1. f ( x) 3 x 5 2. f ( x) f ( x) log 1 x 4 1 x 4 2 2 3. f ( x) log 2 ( x 2) 4. 3 For questions 5-8, solve each word problem. 5. Dustin deposits $1000 into his bank account at 4% annual interest rate. If the account is compounded continuously, how long would it take for Dustin’s account to double? 6. Beth deposits $300 into her bank account at an interest rate of 7%. If the account is compounded weekly, how long would it take for her account to triple? 7. Susan decides to save her money by putting it in a bank account that earns 3% annual interest. Susan puts $2500 in an account whose interest is compounded quarterly. How much money is in Susan’s account after 8 years? 8. Chris deposits $50 into his account that earns 4% annual interest. If the account is compounded continuously how long will it take for Chris to have $75 in his account? For questions 9-14, solve each equation. 9. log 4 x log 4 (2 x - 3) log 4 2 10. 8 105 x 4 35 11. 34 x 813 x 2 12. log( x2 1) log( x 5) 13. ln( x 3) 2 8 14. 2 log(2 x 4) log( x 6) For questions 15-16, expand each expression using your logarithmic properties. 15. log 3 81x 7 y 6 4 16. log 2 16 x8 y 2 ( z 2) z For questions 17-18, condense each expression into a single logarithm using your logarithmic properties. 17. 4log5 x 2log5 y 3log5 z 18. 25log2 m 4log2 n 9log2 p 4log2 k For questions 19-21, evaluate each expression without using your calculator. 1 19. log 1 256 20. log5 625 21. log 7 343 2 16 Math Analysis Notes Chapter 3 Mr. Hayden