Name: Math Analysis Chapter 3 Notes: Eponential and Logarithmic Functions Day : Section 3-1 Eponential Functions 3-1: Eponential Functions After completing section 3-1 you should be able to do the following: 1. Evaluate eponential functions. Graph eponential functions 3. Evaluate functions with base e 4. Use compound interest formulas An eponential functions are functions whose equations contain a variable in the eponent. Definition of the Eponential Function The eponential function f with base b is defined by: Eample of eponential functions: f( ) 3 h ( ) 10 f ( ) b or y b Where b is a positive constant other than 1 (b > 0 and b 0) and is any real number. e ( ) 1 1 k ( ) 3 base of 3 base of 10 base of base of 1/ Eample of functions that are not eponential functions: a ( ) 1 b ( ) c( ) 4 d( ) The base of an eponential function must be a positive constant other than 1. To evaluate epressions with eponents with a calculator: 1. Enter base. find the button [^] or [y ] and push it 3. enter eponent 4. push the equal button and you should have your answer. Practice: Approimate each number using a calculator. Round your answer to three decimal places. 1. 5 3.. The base of an eponential function must be a positive. 6 4 3. Variable is the base and not the eponent. 4 e 4. Variable is both the base and the eponent. 31 6. 1
Graphing Eponential Functions Graphing eponential functions in the form y = ab for b > 1 where a is a real number and b is the base (b 1) Practice: (a) Graph each eponential function. (b) State the domain and range in interval notation. (c) Label the horizontal asymptote. 1. y = 4. y = 1 4 y y Graphing Eponential Functions in the form y = ab h + k You must graph the parent function 1 st. y = ab 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 eponential functions that have translations in them. Practice: (a) Graph each eponential function. (b) State the domain and range in interval notation. (c) Label the horizontal asymptote. 1. y =4 1 3. y = 3 1 y y
The Natural Base e An irrational number, symbolized by the letter e, appears as the base in many applied eponential functions. The number e is defined as the value that 1 1 n n as n gets larger and larger. As n goes to the approimate value of e to nine decimal places is: e.7188187. The irrational number e approimately.7, is called the natural base. The function f ( ) e is called the natural eponential function. Practice: (a) Graph each eponential function. (b) State the domain and range in interval notation. (c) Label the horizontal asymptote. 1. y = e. y = e 3 y 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: r 1. For n compoundings per year: A P1 n nt. For continuous compounding: A Pe rt 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
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. leg opposite to A opp a sin A hypotenuse hyp c B leg adjacent to A adj b cos A hypotenuse hyp c c a leg opposite to A opp a tan A leg adjacent to A adj b 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. cosz 3. tanl 4. sinx 5. cosl 6. tanz L 6 K 6 3 M Z 4 Y 4 X Use for 45-45 -90 Special Right Triangle Use for 30-60 -90 Special Right Triangle 45 30 3 45 60 In 1-6, use the above triangle to find the indicated trigonometric value. 1.) sin30.) cos60 3.) tan 45 4.) cos30 5.) tan 60 6.) sin 45 4
Day 3: 3- Logarithmic Functions 3-: Logarithmic Functions After completing section 3- you should be able to do the following: 1. Change from logarithmic to eponential form.. Change from eponential 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 > 0 and b > 0, b 1: The function y y log b is equivalent to b f ( ) logb 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 eponent. Eponent Eponent Logarithmic Form: y = log b Eponential Form: b y = Base Base To change from logarithmic form to the more familiar eponential form, use this pattern: y = log b means b y = Practice: In 1-4, Write each equation in its equivalent eponential form: 1. 3 = log 7. = log b 5 3. log 4 6 = y 4. log - 3 = y Practice: In 1-4, Write each equation in its equivalent logarithmic form: 1. 5 = 3. b 3 = 7 3. e y = 33 4. 4 = 64 To Evaluate a logarithmic epression without using a calculator: 1. Set logarithmic epression equal to. Write the equation in its equivalent eponential form 3. Evaluate the eponential epression 4. The answer to the eponential epression it the value of the logarithmic epression. Practice: In 1-4, Evaluate each epression without using a calculator. 1 1. log 4 16. log 64 8 3. log 64 4. log4 16 5
Basic Logarithmic Properties Involving One 1. logbb 1 because 1 is the eponent to which b must be raised to obtain b.. logb1 0 because 0 is the eponent to which b must be raised to obtain 1. (b 0 = 1) Inverse Properties of Logarithms For b > 0 and b 1: log b b log b b The logarithm with base b of b raised to a power equal that power b raised to the logarithm with base b of a number equals that number Practice: In 1-4 Evaluate each epression without using a calculator. 1. log 9 9. log 4 1 3. log 7 7 3 4. 8 log 8 19 To Graph a logarithmic Functions in the form y = log b 1. Rewrite the function in eponential form. Graph the eponential equation by making an /y table. You will be choosing values for the eponent (in this case y) 3. Connect points with a smooth curve. Practice: Graph y = and y = log on the same rectangular coordinate system. y = y = log y y Characteristics of graphs of Logarithmic Functions Have vertical asymptotes Domain is restricted by vertical asymptote, however, range is, 6
To Graph a logarithmic Functions in the form y = alog b ( h) + k 1. Write logarithmic function that does not contain transformations h or k.. Write the logarithmic function found in step 1 in eponential form. 3. Follow steps above to graph the eponential function found in step. 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 ( h) + k Practice: (a) graph: y = log 3 ( ) - 4. (b) State the domain and range. (c) Write the equation of the asymptote of the graph. Practice: (a) graph: y = ln( + 1) 3. (b) State the domain and range. (c) Write the equation of the asymptote of the graph. 7
The common base (10): 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 The natural base (e): A logarithm with a base of e is written a natural logarithm. So log e15 is written as ln15 and is read as natural log of 15 Practice: Evaluate or simplify each epression without using a calculator. 1. log100. lne 3. lne 8 4. log10 4 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. hypotenuse hyp c 1 csc A leg opposite to A opp a sin A B hypotenuse hyp c 1 sec A leg adjacent to A adj b cos A c a leg adjacent to A adj b 1 cot A leg opposite to A opp a tan 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. secz 3. cotl 4. cscx 5. secl 6. tanz L 6 K 6 3 M Z 4 Y 4 X 8
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, cos? 8.) tan, sec? 5 1 1 5 9.) cos, csc? 10.) sin,cot? 7 Day 4: Section 3-3 Properties of Logarithms; Section 3-4 Eponential 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. Use the quotient rule 3. Use the power rule 4. Epand logarithmic epressions 5. Condense logarithmic epressions 6. Use the change-or-base property Rules of Logarithms (very similar to the rules of eponents) Let b, M, and N be positive real numbers with b 1. The Product Rule log b (MN) = log b M + log b N The logarithm of a product is the sum of the logarithms. The Quotient Rule log M g log bm log bn N The logarithm of a quotient is the difference of the logarithms. The Power Rule log M log M Practice: 1-4, use the properties of logarithms to epand each logarithmic epression as much as possible. Where possible, evaluate the logarithmic epression without using a calculator. 1. log 7 (7). log 100 b The logarithm of a number with an eponent is the product of the eponent and the logarithm of that number. We use these rules in order to epand or condense a logarithmic epression 3. log 100 4. b log 3 8 1 7 1 9
Practice: 1-3, use the properties of logarithms to condense each logarithmic epression. Write the epression as a single logarithm whose coefficient is 1. 1. log( 5) 3log3. 3(log + logy) (log( + 1)) 3. 4ln + 7lny 3lnz 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 Changing to the Natural Base logm lnm logbm logb M logb lnb Practice: Use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. 1. log 5 13. log 14 87.5 3. log 0.1 1 4. log π 60 3-4: Eponential and Logarithmic Equations After completing section 3-4 you should be able to do the following: 1. Use like bases to solve eponential equations. Use logarithms to solve eponential 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 eponential equations: Method 1: Epressing each side as a power of the same base. Practice: In 1-4, Solve: 1. 5 3 6 = 15. 8 + = 4 3 3. 5 = 1 5 4. 3 4 6 6. 10
Method : Using Natural Logarithms to Solve Eponential Equations Since most eponential equations cannot be rewritten so that each side has the same base. Logarithms are etremely useful in solving such equations. Steps to solve eponential equations using natural logarithms 1. Isolate the eponential epression.. Take the natural logarithm on both sides of the equation. 3. Simplify using one of the following properties: lnb = lnb or lne = 4. Solve for the variable. Practice: In 1-4 Solve: 1. 5 = 134. 7e 5 = 58 3. 3 1 = 7 + 1 4. e 8e + 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.. Use the properties of logarithms to write the epression as a single logarithm whose coefficient is 1. (Condense if necessary) 3. Use the definition of a logarithm to rewrite the equation in eponential form: log b M = c means b c = 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. log ( 4) = 3. 4ln(3) = 8 3. log + log( 3) = 1 4. ln( 3) ln( 7 3) ln( 1) 11
More Trig Review A ratio of the lengths of two sides of a right triangle is called a trigonometric ratio. The si trigonometric ratios are defined for the acute angles of a right triangle as: leg opposite to A opp a sin A hypotenuse hyp c hypotenuse hyp c 1 csc A leg opposite to A opp a sin A B leg adjacent to A adj b cos A hypotenuse hyp c hypotenuse hyp c 1 sec A leg adjacent to A adj b cos A c a leg opposite to A opp a tan A leg adjacent to A adj b leg adjacent to A adj b 1 cot A leg opposite to A opp a tan A A b C A harmonic that can be used to remember the 1 st 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-, Use the given trigonometric equation to find the remaining five trigonometric equations. 4 1.) sin X.) cscg 6 3 Practice: In 3-6, Use the given trigonometric equation to find the indicated trigonometric ratio. 3.) sec X, cot X? 4.) sin X, tan X? 5.) csc X 5, cos X? 6.) 3 tan X, sin X? 1
Day 5: Section 3-5 Eponential Growth and Decay; Modeling Data; Compound Interest Problems 3-5: Eponential Growth and Decay After completing section 3-5 you should be able to do the following: 1. Model eponential growth and decay. Use compound interest formulas to solve word problems Eponential Growth and Decay Models The mathematical model for eponential growth or decay is given by: A = a 0 e kt Where a 0 = 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 Eponential Growth k > 0 Eponential Decay k < 0 Practice: The eponential model A = 106.e.018t describes the population of a country, A, in millions, t years after 003. Use this model to solve Eercises 1-4. 1. What was the population of the country in 003?. Is this county s having a population growth or decay? 3. What will be the population in 01? 4. When will the population be 1000 million? 13
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 P1 n nt For continuous compounding: rt A Pe Practice: 1. Find the number of years it takes for $10,000 to double at an interest rate of 7% compounded quarterly.. Find the number of years it takes $1500 to become $4000 at an interest rate of 5.5% compounded continuously. More Trig Review Remember to use SOH-CAH-TOA & HO-HA-A0 to find the si trigonometric ratios. Hypotenuse Side Opposite to A opp 1 sin A hyp csc A ( SOH ) adj 1 cos A hyp sec A ( CAH ) hyp 1 csc A opp sin A ( HO) hyp 1 sec A adj cos A ( HA) A Side Adj acent to A opp 1 tan A ( TOA) adj cot A adj 1 cot A ( AO) opp tan A Practice: In 1-, Use the given trigonometric equation to find the remaining five trigonometric equations. 5 1.) cos X.) cot G 3 6 14
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? 5.) 5 sec X, sin X? 6.) 3 cot X, sec X? 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. 4 1 1. f( ) 3 5. f( ) 3. f ( ) log ( ) 4. f ( ) log1 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? 15
7. Susan decides to save her money by putting it in a bank account that earns 3% annual interest. Susan puts $500 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. 4 4 4 log log ( - 3) log 10. 54 4 3 810 35 11. 3 81 1. 13. ln( 3) 8 14. log( 1) log( 5) log( 4) log( 6) For questions 15-16, epand each epression using your logarithmic properties. 15. 7 6 81 y log3 4 z 16. 8 log 16 y ( z ) For questions 17-18, condense each epression into a single logarithm using your logarithmic properties. 4log log y 3log z 18. 5log m 4log n 9log p 4log k 17. 5 5 5 For questions 19-1, evaluate each epression without using your calculator. 1 19. log 1 56 0. log5 65 1. log7 343. 3 ln1 1 e + log 48-16
Trig Worksheet #11 Name Period For 1-, Write the si trigonometric functions for angle A. 1.).) C B 6 A 3 1 B 3 A 3 15 C 10 3.) Given sin X what are the values of 13 the 5 other trigonometric ratios? 5 4.) Given tan X what are the values 13 of the 5 other trigonometric ratios? In5-10, Label the sides of both the 30-60-90 and 45-45-90 special right triangles. Then find the indicated trigonometric ratio 5.) sin 30 6.) cos45 45 30 7). tan60 8.) sin60 45 60 9.) cos30 10.) tan45 17
Trig Worksheet #10 Name: Period: In 1-1, Draw the special right triangle to find the indicated trigonometric value. 1.) sin 45.) sec30 3.) cot 60 4.) tan30 5.) csc30 6.) csc45 7.) cot 30 8.) tan 45 9.) sec45 10.) csc60 11.) sec45 1.) sec60 In 13-14, Find the value of the other five trigonometric functions of θ. 15 3 13.) sin 14.) tan 10 5 18
Trig Worksheet #9 Name: Period: 1.) Evaluate the si trigonometric functions of angle θ. 4 3 6 6 In -5, Use the given trigonometric equation to find the indicated ratio. 5 15.) sin, sec? 3.) cos, csc? 7 6 4.) 73 tan, sin? 5.) tan 3, sec? In 6-8, (A) Draw the special right triangle indicated by the problem, (B) then use it to find the indicated trigonometric value. 6.) cot 60? 7.) sin 45? 8.) csc30? 19
Trig Worksheet #8 Use for 45-45 -90 Special Right Triangle Name: Period: Use for 30-60 -90 Special Right Triangle 45 30 3 45 60 In 1-1, use the above triangle to find the indicated trigonometric value. 1.) sin30.) sin 60 3.) sin 45 4.) cos30 5.) cos60 6.) cos45 7.) tan30 8.) tan 60 9.) tan 45 10.) csc30 11.) csc60 1.) csc45 13.) sec30 14.) sec60 15.) sec45 16.) cot 30 17.) cot 60 18.) cot 45 0