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C2D8. Bellwork:. Fill in the blanks on the worksheet. Recall the formula for compound interest:. Initial value (or amount). Number of compoundings. Interest rate as a decimal. Time.
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C2D8 Bellwork: Fill in the blanks on the worksheet Recall the formula for compound interest: 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 . Fill in the table
2 1 4 2.44141 2.61304 12 365 2.71457 8760 2.71813 e 2.718281828 _________ ______________ Now we will use e in logs!!
e 2.718281828 _____ _____________ π This number is a constant (like ___), and is referred to as the _________ base. The logarithm with base e is called the _________ logarithm, ___. natural natural ln
10 8 6 4 2 –10 –6 –4 –2 2 4 6 10 8 –8 –2 –4 –6 –8 –10 Sketch y = f(x) for f(x) = ex. y = x f(x) = ex 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.
Practice #1: exponent a) b) c) argument x = 4 log 5 = 9x log base e e exponent 4 = x log e argument base Never write “e” in the base for the final answer!
Practice #1: d) e) f) 10 = x – 8 16 = 3x + 7 x5 = 2 log log log 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 x x4 3 e = 56 e = 9 e = x – 4 What’s the base?
Properties: Natural logs have the same properties as regular logarithms: Same thing as before except the base is e! Practice #2: 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. Find each to 2 decimal places. 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.
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. Rewrite these with exponents Definition of Logs: + 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? Only x = 6, Sucka! ln x2 = ln (2x + 24) 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? 4 e (exact) x ≈ 36.95 (approx.)
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) = _________ Principal Value P = Compounding rate (as a decimal) r = Time t = 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. 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
