Exponential and Logarithmic Functions! OH YEAH! Unit 5 Exponential and Logarithmic Functions! OH YEAH!

Unit Essential Question How are exponential and logarithmic functions related, and how can they be represented graphically?

Exponential Functions Lesson 5.1 Exponential Functions

Lesson Essential Question (LEQ) What is an exponential function and how can they be represented graphically?

Solving Exponential Equations Examples: 1) 4 3𝑥 = 4 6 2) 5 3𝑥 = 5 7𝑥−8 3) 3 2𝑥−4 = 9 2𝑥+6 4) 2 𝑥 2 −9 = 4 2𝑥+ 3 2

Sketching Graphs Let’s sketch the graph of the following functions: Ex: 𝑓 𝑥 = 3 𝑥 Ex: 𝑓 𝑥 = 3 𝑥−2 Ex: 𝑓 𝑥 = 3 𝑥 +1 Ex: 𝑓 𝑥 = 2 𝑥 Ex: 𝑓 𝑥 = ( 1 2 ) 𝑥

Even More Graphs! Let’s sketch the graph of: Ex: 𝑓 𝑥 = 2 𝑥 2

Real World Applications: Compound Interest Decay/Growth Half-Life Bloodstream Appreciation/Depreciation Inflation Epidemics Many more…

Homework: Page 334-335 #’s 1, 3, 5, 7, 9, 13, 15, 17, 21, 30, 31a, 33a, 35

Bell Work: 1) Solve for x. 8 2𝑥−5 = ( 1 2 ) 𝑥 2 −1

Compound Interest: Blue Table on Page 332 𝐴=𝑃 (1+ 𝑟 𝑛 ) 𝑛𝑡 A = Future Value P = Principal r = interest rate as a decimal n = number of interest periods per year t = number of years Principal is invested

Examples: Ex: If you invested $2,000 dollars 10 years ago at 4.5% that was compounded quarterly, what would be the value of that investment today? Ex: Mr. Kelsey needs to have $2,000,000 by the time he retires in 33 years. He plans to invest in a money market account that will return 5.25% per year. How much money will he need to invest right now to reach his goal?

Classwork/Homework: Pages 335-336 #’s 37 – 42, 45-48

Bell Work: The current average cost of gasoline per gallon in PA is $2.79 and has been increasing at an average inflation rate of 3.75% per year. If this pattern holds true, what will be the cost of gas in 30 years?

The Natural Exponential Function Lesson 5.2 The Natural Exponential Function

LESSON ESSENTIAL QUESTION What is the natural exponential function and how can it be used?

Important: (1+ 1 𝑛 ) 𝑛 =𝑒=2.71828…𝑎𝑠 𝑛 𝑎𝑝𝑝𝑟𝑜𝑎𝑐ℎ𝑒𝑠 ∞ (1+ 1 𝑛 ) 𝑛 =𝑒=2.71828…𝑎𝑠 𝑛 𝑎𝑝𝑝𝑟𝑜𝑎𝑐ℎ𝑒𝑠 ∞ How do we get this???? The NATURAL EXPONENTIAL FUNCTION: 𝑓 𝑥 = 𝑒 𝑥

Continuously Compounded Interest 𝐴=𝑃 𝑒 𝑟𝑡 A = Future Value P = Principal r = interest rate as a decimal t = number of years Principal is invested

Law of Growth/Decay Formula 𝑞 𝑡 = 𝑞 0 𝑒 𝑟𝑡 𝑞 𝑡 =𝑄𝑢𝑎𝑛𝑡𝑖𝑡𝑦 𝑞 0 =𝐼𝑛𝑖𝑡𝑖𝑎𝑙 𝐴𝑚𝑜𝑢𝑛𝑡 𝑜𝑓 𝑞 𝑟=𝑟𝑎𝑡𝑒 𝑜𝑓 𝑔𝑟𝑜𝑤𝑡ℎ 𝑜𝑟 𝑑𝑒𝑐𝑎𝑦 𝑎𝑠 𝑎 𝑑𝑒𝑐𝑖𝑚𝑎𝑙 𝑡=𝑡𝑖𝑚𝑒 𝑖𝑛 𝑦𝑒𝑎𝑟𝑠 If r > 0, then the quantity is growing. If r < 0, then the quantity is decaying.

Homework: Pages 345-346 #’s 5 – 15 odds, 19-31 odds

Bell Work: 1) If $12,000 is continuously compounded at 2.55% over 5 years and 3 months, what would be the amount of interest earned? 2) The population of PA in 1990 was 12 million. If the population is continuously growing at a rate of 1.05% per year, then what would the population be in 2020? 3) The amount A(t) of a certain radioactive isotope after t days is given as 𝐴 𝑡 = 𝐴 𝑜 𝑒 𝑟𝑡 where 𝐴 𝑜 is the initial amount of the isotope and the rate of decay is 0.225%. What was the initial amount if after 50 days, there was 893.6 grams left?

Small Quiz Monday: You will need to know for tomorrow: Solve exponential equations Compound Interest Formula Word Problems Solve natural exponential equations Continuously Compounded Interest Growth/Decay Formula Word Problems! WORD PROBLEMS!!!!!

Classwork/Homework: Pages 345-347 #’s 6, 8, 12, 14, 20, 22, 24, 26a, 28, 32 This assignment will be collected!

Bell Work: 1) Solve: 9 2𝑥 ∙ ( 1 3 ) −5𝑥 = 27 −4 ∙ ( 1 9 ) −1 3) The half-life of a radioactive isotope is 400 years. If there are 600 mg of the isotope present at time t = 0, then the amount remaining after t years is given as 𝐴 𝑡 =600 (2) −𝑡/400 . A) How much of the isotope is remaining after 200 years? B) Using the given equation, how long will it take for 75 mg of the isotope to be left? C) Explain how you know the amount remaining after 2000 years without using the equation.

Examples: 4) Mackenzie borrows $20,000 to purchase a new car. Her loan is to be compounded monthly over 5 years at 4.5%. What will be her monthly payments? 5) The local population of koala bears has been decreasing since 1975 at a continuous rate of 6%. If the population is currently estimated to be 100 (in 2015), what would be the estimated population back in 1975? 6) Solve: 64 2𝑥−5 = ( 1 32 ) −6

Bell Work: 1) Colten begins a new job working for $15.25 an hour. Every year he will get a annual 3% cost of living raise. If he works at this same job for 30 years, what will be his hourly wage?

Logarithmic Functions Lesson 5.3 Logarithmic Functions

Lesson Essential Question: How are logarithmic functions related to exponential functions and what are the different properties of logarithms?

Logarithm 𝑦= log 𝑎 𝑥 if and only if 𝑥= 𝑎 𝑦 x must be > 0 y must be a real number How does this compare to an exponential function?

Graphs of Logarithms Let’s sketch the graph of a basic logarithmic function and see how it compares to a basic exponential function. Sketch: 𝑓 𝑥 = log 2 𝑥 and 𝑔 𝑥 = 2 𝑥

Rewriting Logarithms We can rewrite any logarithm in exponential form. Let’s rewrite the following logarithms as exponents: log 2 16 =𝑥 log 𝑦 625 =4 log 10 𝑚 =3

Rewriting Exponents We can also rewrite exponents as logarithms.

Simplifying Logarithms Simplify (if possible): Ex: log 3 1 81 Ex: log 2 64 Ex: log 12 1 Ex: log 5 −25

Bell Work: Simplify: 1) log 6 1 36 = ? 2) log 2 16 = ? 3) log 8 2 = ?

Properties of Logarithms Page 350 Blue Table Remember These Properties!!!!!

Solving Logarithmic Equations Ex: log 4 5+𝑥 =3 Ex: log 5 𝑥 2 −11 =2 Ex: log 3 (𝑥−8) = log 3 ( 𝑥 2 −14) Ex: log 𝑥 (− 5𝑥 2 +9𝑥+45)=3

Common Logarithm The most basic form of a logarithm: log 𝑥= log 10 𝑥 If the log does not have a specified base, it is assumed to be a base 10 log. Your calculator will only do base 10 logarithms!

NATURAL LOGARITHMS!!!!!!!!! Just like there was a natural exponential function “e”, we also have a natural logarithmic function! YES! ln 𝑥 = log 𝑒 𝑥 𝑎𝑠 𝑙𝑜𝑛𝑔 𝑎𝑠 𝑥>0 Just like exponential functions and logarithms are inverses, the natural exponent and natural log are inverses!!! Ex: ln 𝑥 =2 𝑐𝑎𝑛 𝑏𝑒 𝑟𝑒𝑤𝑟𝑖𝑡𝑡𝑒𝑛 𝑎𝑠 𝑒 2 =𝑥

Blue Table on Page 356 These are four properties of logarithms you should know, as well as: If 𝑦=𝑎 𝑏 𝑥 , then 𝑦=𝑎 𝑒 𝑥∙ ln 𝑏 Ex: Convert 5∙3 𝑥 to a base e expression.

Homework: Pages 359 – 360 #’s 2, 4, 10, 12, 14, 16 – 32 evens

Bell Work: Be ready to ask questions on the homework! If not, begin working on the following assignment which is due Monday! (It will be collected!) Pages 359 – 360 #’s 1, 3, 13 – 31 odds

Bell Work: Solve: 1) log 9 27 =𝑥 2) 5𝑒 ln 𝑥 =45 3) 𝑒 𝑥 ln 5 =0.04 4) Solve for t. 100=200 𝑒 −0.05𝑡 5) Solve for k. 40=160( 10) 1.5𝑘

Graphing Logarithms Lets create a table and sketch the graphs of the following: 𝑦= log 3 𝑥 𝑦= log 2 𝑥 𝑦= ln 𝑥 (graphing calculator) What do you notice about these graphs compared to exponential functions?

Shifting/Reflecting The graphs of logarithms behave just like any other functions. Lets sketch the graphs of some functions that will shift and/or reflect.

Class Examples: Pages 362 – 363 #’s 52, 54, 60, 66, 70

Homework: Page 360 #’s 51 - 69 odds

Bell Work: The current population of Bloomsburg in 2015 is approximately 12,000. It is projected to grow continuously in the future at a rate of 1.85%. How long will it take for the population of Bloomsburg to double in size?

Properties of Logarithms Lesson 5.4 Properties of Logarithms

Lesson Essential Question What are the different properties of logarithms and how are they used when simplifying exponential and/or logarithmic expressions?

Three More Properties of Logarithms ORANGE TABLE ON PAGE 364 These properties hold true for the common logarithm and the natural logarithm. (Blue Table Page 365)

Examples: Using the laws of logarithms, rewrite the expression using logs of x, y, or z. Ex: log 𝑎 𝑥 5 𝑦 Ex: log 𝑎 𝑥 4 𝑦 10 Ex: ln 3 𝑥 4 𝑦 2 𝑧 3

Examples: Using the properties of logarithms, rewrite each expression as one logarithm. Ex: 1 3 log 𝑎 𝑥 −2 log 𝑎 𝑦 Ex: 1 3 log 𝑎 ( 𝑥 2 −1) −3 log 𝑎 𝑦 − log 𝑎 𝑧

Solving Logarithmic Equations: Solve each logarithmic equation. Double check to make sure the solution you found is in fact a solution! Ex: log 7 (8𝑥−12) = log 7 2𝑥 + log 7 5 Ex: log 4 𝑥 + log 4 (𝑥+4) = log 4 12 Ex: 2 ln 𝑥−2 − ln 4 = ln 25 Ex: 3 ln 𝑥 = ln 𝑒 3 + ln 8

Homework: Page 370 #’s 1-33 odds

Bell Work: Get out your homework from last night: Page 370 1-33 odds Be ready to ask questions!

Quiz Tomorrow: LOGARITHMS! (remember that to answer some logarithmic problems you need to know how to change to exponential form) On the quiz: Solving Logarithms Word Problems Properties of Logarithms WORD PROBLEMS!!!

Class Work: Pages 370 – 371 #’s 6, 8, 10, 12, 14, 18, 20, 22, 26, 34, 51, 52, 53, 54, 56

Bell Work: 1) Armaan invests $25,000 into a mutual fund that has a continuously compounded interest rate of 4.75%. How long will it take for Armaan to triple his investment? 2) Solve: ln 25= ln 5+ ln 1−0.4ℎ 3) Solve: 2 log (𝑥−5) − log 2 = log 32

Exponential and Logarithmic Equations Lesson 5.5 Exponential and Logarithmic Equations

Lesson Essential Question How can we change the bases of logarithmic and exponential functions, and how do we use the special base formulas?

Example: Solve: 4 𝑥 =20 We can rewrite this in logarithmic form, but we still can’t solve it. Or can we?????????

Change of Base Formula: log 𝑏 𝑢 = log 𝑎 𝑢 log 𝑎 𝑏 We can rewrite a logarithm with a “difficult” base as a quotient of two common logarithms or two natural logs. Ex: log 5 10 = log 10 log 5 = ln 10 ln 5 We can prove it!

Examples: Solve each equation for x first, then approximate each to two decimal places: Ex: 3 𝑥 =18 Ex: 4 2𝑥−5 =10 Ex: 5 2𝑥+1 = 6 𝑥−2

Solving an Exponential Equation This problem is off the hook yo!!!!!!!!!!!! Ex: Solve for x, then approximate to the nearest 2 decimal places: 5 𝑥 − 5 −𝑥 2 =3

Homework: Pages 381-382 #’s 1-31 odds only

Bell Work: Solve for x. 1= 10 𝑥 + 10 −𝑥 2 5= 6 𝑥 + 6 −𝑥 3

Solving a Logarithmic Equation: This pizzle is fo rizzle ( LOL): Ex: Solve: log 3 𝑥 = log 𝑥

Finding an inverse function: Same type of problem that we solved in the bell work. Example: Find the inverse function of 𝑦= 2 𝑒 𝑥 + 𝑒 −𝑥 How could we use our graphing calculator to prove that the functions are indeed inverses?

Example using the Logistic Curve A logistic curve is the graph of an equation in the form: 𝑦= 𝑘 1+𝑏 𝑒 −𝑐𝑥 , where b, c, and k are constants, x represents the time, and y is the population. Example: Assume c = 1.1244, k = 105, and x will be the time in days. A) Find the value of b if the initial population was 3. B) How long will it take the population to reach 90? C) Show that after a long period of time, the population of this curve will become the constant k.

Class Work/Homework: Pages 381-383 #’s 33 – 39 odds 49, 53b, 54, 55, 56a, 56c, 57 If you need extra practice, try the evens (2 – 40)

Unit Test Thursday and Friday we will be having our Unit 5 Test on exponential and logarithmic functions. Here is a group of review exercises to try: Pages 385 – 387 #’s 17 – 40, 45 and 46 (just find the inverse), 47 – 55, 58, 61, 62, 66, 67