W ARM UP 1. Eric and Sonja are determining the worth of a $550 investment after 12 years in an account with 3.5% interest compounded monthly. Eric thinks the investment is worth $837.08, while Sonja thinks it is worth $ Are either of them right? 2. A certain plant has a doubling time of 15 days. If there are 58 plants in a field to begin with, how many will there be after a week? How about a month? (With 30 days in it.)

QUIZ! Take 3 minutes to review your foldable from yesterday, then get out a blank piece of notebook paper and put your name at the top.

QUIZ! Write the formulas in the following order: (You do NOT have to write the names) 1. Half Life 2. Continuous Compound Interest 3. Exponential Growth/Decay 4. Doubling Time 5. Continuous Exponential Growth/Decay 6. Compound Interest BONUS: What is the name of the mathematician who discovered “e”?

I will put up a multiple choice problem. You will work out the problem on a scratch sheet of paper and then hold up the color card that matches the answer choice you pick. Green = A Blue = B Yellow = C Orange = D

Daisy invested $1000 in a savings account at Wells Fargo that pays her 5% interest compounded monthly. If she leaves her money in the account for 20 years, how much will she have? A. $ B. $ C. $ D. $

Under the right growing conditions, a weed has a doubling time of 12 days. Suppose Mrs. Inscoe’s front yard contains 46 weeds of this species, and she’s too busy to get rid of them right now. How many weeds will be there after 20 days? A. 14 weeds B. 146 weeds C. 70 weeds D. 200 weeds

The pressure of the atmosphere at seal level is pounds per square inch. It decreases continuously at a rate of 0.004% as altitude increases by x feet. What will the atmospheric pressure be at the top of Mount Everest which is 29,035 feet above sea level. Hint: In this problem, “t” doesn’t represent time, but rather feet above sea level. A B C D. 4.76

Unique has saved up $1500 that she plans to put in a savings account at Bank of America that pays 5% interest. How much will she have after 10 years if the interest is compounded continuously? A. $ B. $ C. $7, D. $

The population of North Carolina is growing at a rate of 1.02% each year. If North Carolina currently has 9 million residents, how many people will live in NC in 2020? A million B million C million D million

The half-life of a radioactive substance is the amount of time it takes for half of the atoms in it to disintegrate. Uranium-235 is used to fuel Duke Power’s big power plant, and has a half life of 704 million years. If Duke power starts with 200 grams, how many grams will be left after 1 million years? A g B. 0 g C g D g

B UT WHAT ABOUT THIS ? I put $200 in the bank several years ago, and now I have $300. The bank paid me interest at a rate of 3% compounded continuously. For how long did I leave my money in the bank? How would I set this up? How would I solve it?

L OGARITHMS VS. EXPONENTS Logarithm: the inverse of an exponent In other words: If f(x) = b x, then f -1 (x) = log b x

B UT WAIT … DOES AN EXPONENTIAL FUNCTION HAVE AN INVERSE ? How do we test a function to determine whether or not it has an inverse? (unit 2 anyone??)

F( X ) = 3 X Domain: -∞, ∞ Range: 0, ∞ Intercepts: no x, y = 1 Asymptotes: x-axis (y=0) End behavior: x F(x) 1/91/ TO FIND THE INVERSE, WE SWITCH X and Y!

F ( X ) = LOG 3 X Domain: Range: Intercepts: Asymptotes: End behavior: x 1/91/ F(x)

F ( X ) = 3 X F ( X ) = LOG 3 X

F ( X ) = LOG 1/2 ( X + 2) What transformation happened? Domain: Range: Intercepts: Asymptotes: End behavior: x 11/21/41/81/161/32 F(x)

R ECAP Log functions are the inverse of exponential functions To find the inverse, you switch x and y. That means you switch the domain and range, x- intercepts and y-intercepts, and horizontal and vertical asymptotes.

T HINK P AIR S HARE Work with your neighbors to compare and contrast logarithmic and exponential equations and graphs.

S O WHAT ARE LOGARITHMS ANYWAYS ? We know they are the inverse of of exponents, but how do they work? Let’s practice going back and forth between logarithmic and exponential form to see.

C AN YOU FIGURE IT OUT ? log 5 25 = 2 log 4 64 = 3 log = 4 log 7 7 = 1

W RITE THESE IN EXPONENTIAL FORM log = 2 log = 3 log = 5

NOTE Common Logarithm: log 10 (log with a base 10) is called a common logarithm and is often written without a base. So, if you see log50, assume that it means log Natural Logarithm: log e (log with a base e) is called a natural logarithm, and is usually written ln So, if you see ln50, assume that it means log e 50

E VALUATE LOGARITHMIC EXPRESSIONS We use the skill we just practiced to evaluate logarithmic expressions. Example: Evaluate log 6 36 Set it equal to x Rewrite as an exponential equation Determine x

T RY THESE TWO : log100000log 9 81

P RACTICE P ROBLEMS Complete the problems on the back of your guided notes sheet for today. Working diligently earns you your class work stamp! HOMEWORK: complete problems 23, 25ab, 37b, 40ab from yesterday’s notes and STUDY FOR YOUR QUIZ MONDAY!

P APERWORK I have lots of papers to return to you today— unfortunately, your tests will not come back to you until Monday. I need YOU to give me your tracking sheet and exit ticket in my inbox on the way out the door.

E XIT TICKET 1. Rewrite the expression in exponential form: log Evaluate the following expressions: log1000 log 2 ½ 3. Determine the domain, range, x-intercept, and vertical asymptote of y = ln(x – 3)