C2D8 Bellwork: Fill in the table Fill in the blanks on the worksheet Recall the formula for compound interest: P = 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 P = 1, r = 100% = 1, and t = 1. Let n  . n A(t) Yearly n = 1 2 Quarterly n = Fill in the table

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 2.44141 12 2.61304 365 2.71457 8760 2.71813 e 2.718281828 _________  ______________ Now we will use e in logs!!

e _____  _____________ 2.718281828 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

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) =

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.

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!

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

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?

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)

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

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.

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

 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

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

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

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).

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 ≈ 13.73 years (approx.)

b) An initial deposit of $2000 is now worth $5000 b) An initial deposit of $2000 is now worth $5000. 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