3.1 Exponential Functions – any function whose equation contains a variable in the exponent. [measures rapid increase or decrease (Example: epidemic growth)] f(x) = bx f – exponential function b - constant base (b > 0, b  1) x = any real number -- domain is (-∞ , ∞) f(x) = 2x g(x) = 10x h(x) = 3x+1 Shift up c units f(x) = bx + c Shift down c units f(x) = bx – c Shift left c units f(x) = bx+c Shift right c units f(x) = bx-c f (x) = 3x g(x) = 3x+1 (0, 1) (-1, 1) 1 2 3 4 5 6 -5 -4 -3 -2 -1 Graphing Exponential Functions: 32+1 = 33 = 27 32 = 9 2 31+1 = 32 = 9 31 = 3 1 30+1 = 31 = 3 30 = 1 3-1+1 = 30 = 1 3-1 = 1/3 -1 3-2+1 = 3-1 = 1/3 3-2 = 1/9 -2 g(x) = 3x+1 f (x) = 3x x

See P.193 All properties above are the same Except #4 which is ‘decreasing’

3.2 Investment and Interest Examples Compound Interest Karen Estes just received an inheritance of $10000 And plans to place all money in a savings account That pays 5% compounded quarterly to help her son Go to college in 3 years. How much money will be In the account in 3 years? Use the formula: A = P(1 + r/n)nt Simple Interest Juanita deposited $8000 in a bank for 5 years at a simple interest rate of 6% 1. How much interest will she get? 2. How much money will she have at the end of 5 years Use the formula: I = Prt I = (8000)(.06)(5) = $2400 (interest earned in 5 years) In 5 years she will have a total of A = P + I = 8000 + 2400 = $10,400 A = amount in account after t years P = principal or amount invested t = time in years r = annual rate of interest n = number of times compounded per year Annual => n = 1 Semiannual => n = 2 Quarterly => n = 4 Monthly => n = 12 Daily => n = 365 As n increases (n  ∞), A does NOT increase indefinitely. It gets closer and closer to a fixed number “e”, and gives us the continuous compound interest formula: A = Pert

Compound Interest related to Simple Interest If a principal of P dollars is borrowed for a period of t years at a perannum interest rate r, the interest I charged is: I = Prt Annually -> Once per year Semiannually -> Twice per year Quarterly -> Four times per year Monthly -> 12 times per year Daily -> 365 times per year Example: A credit union pays interest of 8% per annum compounded quarterly on a Certain savings plan. If $1000 is deposited in such a plan and the interest is Left to accumulate, how much is in the account after 1 year? I = (1000)(.08)(1/4) = $20 => New principal now is: $1020 I = (1020)(.08)(1/4) = $20.40 => New principal now is: $1040.40 I = (1040.40)(.08)(1/4) = $20.81 => New principal now is: $1061.21 I = (1061.21)(.08)(1/4) = $21.22 => New principal now is: $1082.43 Compound Interest Formula: The amount, A, after t years due to a principal P Invested at an annual interest rate r compounded n times per year is: A = P  (1 + r/n)nt

The Natural Base e An irrational number, symbolized by the letter e, appears as the base in many applied exponential functions. This irrational number is approximately equal to 2.72. More accurately, The number e is called the natural base. The function f (x) = ex is called the natural exponential function. Analogous to the continuous compound interest formula, A = Pert The model for continuous growth and decay is: A(t) = A0ekt A(t) = the amount at time t A0 = A(0), the initial amount k = rate of growth (k>0) or decay (k < 0) In year 2000, population of the world was about 6 billion; annual rate of growth 2.1%. Estimate the population in 2030 and 1990. 2000=year 0, 2030=year 30, year 1990=year -10 A(30) = 6e(.021)(30) = 11.265663 (billion people) A(-10) = 6e(.021)(-10) = 4.8635055 (billion people) ** Note these are just estimates. The actual population in 1990 was 5.28 billion f (x) = ex f (x) = 2x f (x) = 3x (0, 1) (1, 2) 1 2 3 4 (1, e) (1, 3) -1

Exponential Growth & Decay (Law of Uninhibited Growth/Decay) A0 is the original amount of ‘substance’ (drug, cells, bacteria, people, etc…)

Solving Exponential Equations e2x-1 = e-4x e3x e2x-1 = e-7x 2x – 1 = -7x -1 = -9x X = 1/9

3.3 Logarithmic Functions A logarithm is an exponent such that for b > 0, b  1 and x > 0 y = logb x if and only if by = x Logarithmic equations Corresponding exponential forms 2 = log5 x 1) 52 = x 3 = logb 64 2) b3 = 64 log3 7 = y 3) 3y = 7 y = loge 9 4) ey = 9 log25 5 = 1/2 because 251/2 = 5. 25 to what power is 5? log25 5 log3 9 = 2 because 32 = 9. 3 to what power is 9? log3 9 log2 16 = 4 because 24 = 16. 2 to what power is 16? log2 16 Logarithmic Expression Evaluated Question Needed for Evaluation Evaluate the Logarithmic Expression

Logarithmic Properties Logb b = 1 1 is the exponent to which b must be raised to obtain b. (b1 = b). Logb 1 = 0 0 is the exponent to which b must be raised to obtain 1. (b0 = 1). logb bx = x The logarithm with base b of b raised to a power equals that power. b logb x = x b raised to the logarithm with base b of a number equals that number. Graphs of f (x) = 2x and g(x) = log2 x [Logarithm is the inverse of the exponential function] 4 2 8 1 1/2 1/4 f (x) = 2x 3 -1 -2 x 2 4 3 1 -1 -2 g(x) = log2 x 8 1/2 1/4 x Reverse coordinates. Properties of f(x) = logb x Domain = (0, ∞) Range = (-∞, +∞) X intercept = 1 ; No y-intercept Vertical asymptote on y-axis Decreasing on 0<b<1; increasing if b>1 Contains points: (1, 0), (b, 1), (1/b, -1) Graph is smooth and continuous -2 -1 6 2 3 4 5 f (x) = 2x f (x) = log2 x y = x

Common Logs and Natural Logs A logarithm with a base of 10 is a ‘common log’ log10 1000 = ______ because 103 = 1000 If a log is written with no base it is assumed to be 10. log 1000 = log10 1000 = 3 3 A logarithm with a base of e is a ‘natural log’ loge 1 = ______ because e0 = 1 If a log is written as ‘ln’ instead of ‘log’ it is a natural log ln 1 = loge 1 = 0

Doubling Your Investments How long will it take to double your Money if it earns 6.5% compounded continuously? Recall the continuous compound interest formula: A = Pert If P is the principal and we want P to double, the amount A will be 2P. 2P = Pe(.065)t 2 = e(.065)t Ln 2 = ln e(.065)t Ln 2 = .065t T = ln 2 = .6931 ≈ 10.66 ≈ 11 years .065 .065 At what rate of return compounded continuously would your money double in 5 years? A = Pert 2P = Per(5) 2 = er(5) ln 2 = ln e5r .6931 = 5r r = .6931/5 R = .1386 So, annual interest rate needed is: 13.86%

Newton’s Law of Cooling T = Ts + (T0 – Ts)e-kt T = Temperature of object a time t Ts = surrounding temperature T0 = T at t = 0 A McDonald’s franchise discovered that when coffee is poured from a CoffeeMaker whose contents are 180° F into a noninsulated pot, after 1 minute, the Coffee cools to 165° F if the room temperature is 72° F. How long should employees wait before pouring coffee from this noninsulated pot into cups to deliver it to customers at 125°F? T0 = 180 and Ts = 72 Now, T = 72 + 108e-.1495t T = 72 + (180 – 72) e-kt 125 = 72 + 108e-.1495t T = 72 + 108e-kt 53 = 108e-.1495t .4907 = e-.1495t Since T = 165 when t = 1 ln .4907 = ln e-.1495t 165 = 72 + 108e-k(1) -.7119 = -.1495t 93 = 108e-k t = 4.76 93 = 108e-k Employees should wait about 5 min Ln .8611 = ln e-k to deliver the coffee at the desired -.1495 = -k temperature. K = .1495

3.4 Properties & Rules of Logarithms Basic Properties Logb b = 1 1 is the exponent to which b must be raised to obtain b. (b1 = b). Logb 1 = 0 0 is the exponent to which b must be raised to obtain 1. (b0 = 1). Inverse Properties logb bx = x The logarithm with base b of b raised to a power equals that power. b logb x = x b raised to the logarithm with base b of a number equals that number. For M>0 and N > 0 Product Rule logb(MN) = logb M + logb N Quotient Rule logb M = logb M - logb N N Power Rule p logb M = p logb M

Logarithmic Property Practice Quotient Rule Product Rule logb M = logb M - logb N N logb(MN) = logb M + logb N log3 (27 • 81) = 2) log (100x) = 3) Ln (7x) = 1) log8 23 x 2) Ln e5 11 = = Power Rule log5 74 Log (4x)5 Ln x2 = Ln x = logb M = p logb M p = =

Expanding Logarithms logb(MN) = logb M + logb N logb M = logb M - logb N N logb M = p logb M p 1) Logb (x2 y ) 3) log5 x 25y3 4) log2 5x2 3 2) log6 3 x 36y4

Condensing Logarithms logb(MN) = logb M + logb N logb M = logb M - logb N N logb M = p logb M p Note: Logarithm coefficients Must be 1 to condense. (Use power rule 1st) log4 2 + log4 32 4) 2 ln x + ln (x + 1) Log 25 + log 4 5) 2 log (x – 3) – log x Log (7x + 6) – log x 6) ¼ logb x – 2 logb 5 – 10 logb y

The Change-of-Base Property Example: Evaluate log3 7 Most calculators only use: Common Log [LOG] (base 10) Natural Log [LN] (base e) It is necessary to use the change Of base property to convert to A base the calculator can use.

Matching Data to an Exponential Curve Find the exponential function of the form f(x) = aebx that passes through the points (0,2) and (3,8) F(0) = 2 f(3) = 8 2 = aeb(0) 8 = aeb(3) f(x) = 2e((ln4)/3)x 2 = ae0 8 = 2e3b 2 = a(1) 4 = e3b a = 2 ln 4 = ln e3b ln 4 = 3b b = (ln4)/3

3.5 Exponential & Logarithmic Equations Step 1: Isolate the exponential expression. Step 2: Take the natural logarithm on both sides of the equation. Step 3: Simplify using one of the following properties: ln bx = x ln b or ln ex = x. Step 4: Solve for the variable using proper algebraic rules. Examples (Solve for x): log4(x + 3) = 2. log 2 (3x-1) = 18 54x – 7 – 3 = 10 3x+2-7 = 27

More Equations to Try