Presentation is loading. Please wait.

Presentation is loading. Please wait.

U6D12 Have out: Bellwork: Fill in the table

Similar presentations


Presentation on theme: "U6D12 Have out: Bellwork: Fill in the table"— Presentation transcript:

1 U6D12 Have out: Bellwork: Fill in the table
Assignment, red pen, pencil, highlighter, GP notebook, calculator U6D12 Have out: Bellwork: Fill in the blanks on the worksheet Recall the formula for compound interest: A0 = n = r = t = Initial value (or amount) Number of compoundings Interest rate as a decimal Time To develop an equation to determine continuously compounded interest, let A0 = 1, r = 100% = 1, and t = 1. Let n  n A(t) Yearly n = 1 2 Quarterly n = Fill in the table

2 e 2.718281828 Now we will use e in logs!! 2 1 4 2.44141 12 2.61304 365
A(t) Yearly n = Quarterly Monthly Daily Hourly 2 1 4 12 365 8760 e _________  ______________ Now we will use e in logs!!

3 e _____  _____________ This number is a constant (like ___), and is referred to as the _________ base. The logarithm with base e is called the _________ logarithm, ___. π natural natural ln

4 ln x Sketch y = f(x) for f(x) = ex. f(x) e-1 ≈ 0.37 e0 = 1 ≈ 2.72 e1
y = x x f(x) (exact) (approximate) –1 1 2 f(x) = ex 4 6 8 10 2 –2 –4 –6 –8 –10 y e-1 ≈ 0.37 y = f –1(x) e0 = 1 x ≈ 2.72 e1 ≈ 7.39 e2 Sketch y = f–1(x) and y = x. ln x f–1(x) =

5 Definition: Whenever you convert back and forth between exponential and logarithmic form using base e, follow the same definition of logs: Why do base e and natural logs exist? In the “real” world, you will see examples of both in situations involving growth and decay of natural organisms, population growth, continuous compounding, etc.

6 Never write “e” in the base for the final answer!
Practice #1: 1. Rewrite each equation in logarithmic form. exponent a) b) c) argument log x = 4 log 5 = 9x base e e exponent log 4 = x e base argument Never write “e” in the base for the final answer!

7 Practice #1: 1. Rewrite each equation in logarithmic form. d) e) f) log 10 = x – 8 log 16 = 3x + 7 log x5 = 2 e e e

8 2. Rewrite each equation in exponential form.
a) ln x = 2 b) ln 3 = x c) ln x2 = 2 2 x 2 e = x e = 3 e = x2 d) ln (x – 4) = 3 e) x = ln 56 f) ln 9 = x4 3 x x4 e = x – 4 e = 56 e = 9 What’s the base?

9 Same thing as before except the base is e!
Properties: Natural logs have the same properties as regular logarithms: Property: Logs Natural Logs Product Quotient Power Practice #2: Same thing as before except the base is e! 3. Condense each expression. a) b)

10 4. Expand the expression. a) b) 5. Simplify each expression. a) ln e b) ln e2 c) ln e x + 4 d) eln 5 e) eln (x – 1) (x + 4) ln e logee 2 ln e x – 1 5 e to what power is e? 2 (1) x + 4 Remember it’s as if the bases “cancel” 2 1

11 Change Of Base: The change of base formula also works for natural logs
Change Of Base: The change of base formula also works for natural logs. Find each to 2 decimal places. Using base 10 Using base e 30 30 1) 2.45 2.45 4 4 2) Sometimes there are rounding errors, so ALWAYS do the entire problem in the calculator, then approximate at the end.

12 Rewrite these with exponents
Practice #3: Solve for the variable. Be careful! All types of logarithmic equations are included below, but you will work with natural logs. Check your solutions.  Definition of Logs: Rewrite these with exponents + 5 + 5 ln (2x + 3) = 3 3 2x + 3 = e Exact and approximate answers are both required! – 3 – 3 2x = e3 – 3 (exact) (approx.) x ≈ 8.54

13  Property of Equality: ln ( ) = ln ( )
Set each argument equal Did you check? ln x2 = ln (2x + 24) Only x = 6, Sucka! x2 = 2x + 24 x2 – 2x – 24 = 0 (x – 6)(x + 4) = 0 x – 6 = 0 or x + 4 = 0 x = 6 x = –4

14  Log Properties: Product, Quotient, and Power
Simplify using the properties Did you check? Only x = 9 Sucka!

15 Combine, then rewrite.  ln( ) + ln( ) = # Did you check? e x ≈ 36.95
4 e (exact) x ≈ 36.95 (approx.)

16 If you see an “e”, use “ln”
 Logging both sides a) If you see an “e”, use “ln” (exact) (approx.) x ≈ –2.91 Be careful here… “ln” before subtracting. Do not write ln(16).

17 Continuously Compounded Interest Formula:
Value after continuous compounding A(t) = A(t) = _________ P = Principal Value r = Compounding rate (as a decimal) t = Time a) An investment of $100 is now valued at $300. The annual interest rate is 8% compounded continuously. About how long has the money been invested? A(t) = P = r = t = 300 100 0.08 100 100 0.08 0.08 (exact) What do we do now? t ≈ years (approx.)

18 Finish today's assignment:
Worksheets

19 b) An initial deposit of $2000 is now worth $5000
b) An initial deposit of $2000 is now worth $ The account earns 5% annual interest, compounded continuously. Determine how long the money has been in the account. A(t) = P = r = t = 2000 2000 0.05 0.05 18.33 years


Download ppt "U6D12 Have out: Bellwork: Fill in the table"

Similar presentations


Ads by Google