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Math 142: Lecture 3-Exponential model and Logarithmic Function Yanfang Yang June 4th, 2014 1 exponential function Definition 1.1. An general exponential function has the form f (x) = abx where a is a real constant number with a 6= 0 and the base, b, is a real constant number with b > 0 and b 6= 1. Remark 1.1. Properties of General Exponential Functions: • Domain: (−∞, ∞) • All graphs are continuous curves. • y = 0 is a horizontal asymptote in one direction. • y-intercept: (0, a) • If a > 0 and b > 1, then the graph increases as x increases (exponential growth function). • If a > 0 and 0 < b < 1, then the graph decreases as x increases (exponential decay function). Proposition 1.1. Exponent Laws and Properties of Exponential Functions: For a and b positive, a 6= 1, b 6= 1, and x and y real, • ax ay = ax+y ; • ax ay = ax−y ; • ax y = axy ; • (ab)x = ax bx ; • ( ab )x = ax bx ; • ax = ay if and only if x = y; • for x 6= 0, ax = bx if and only if a = b. √ Example 1.1. Find the domain of f (x) = π x . 1 Example 1.2. Simplify x3 y 2 z (y 3 z 2 )4 ; 2 Solve for x: 7x = 7−2x+8 . Solve for x: x2 ex − 5xex = 0. Example 1.3. Classify the following functions as exponential growth or decay functions. • f (x) = 5.5(0.5)x • g(x) = 0.25(3.4)x • h(x) = 2ex 2 exponential model Modeling population growth and radioactive decay often uses an exponential function of the form y(t) = Cekt , where t represents time, C represents the initial population (or amount), and k is the relative growth rate constant. e is a constant which is approximately equal to 2.71828. We call e the natural number. If k is positive, it is exponential growth model, if k is negative, it is exponential decay model. Example 2.1. : In 2006, the estimated population in Mathland was 75 million people with a relative growth rate of 2.3%. (a) Write an equation that models the population growth in Mathland, letting 2006 be year 0. (b) Based on the model, what is the expected population in Mathland (to the nearest million) in 2015? In 2030? 2 Example 2.2. The population of an undesirable city is modeled by p(t) = ce−0.08t where t represents the number of years since 2000, p(t) represents the population in the tth year, and c is a constant representing the population in 2000. (a) If the city had a population of 10,000 in 2050, what is the value of c? (b)Use the model to predict the population in 2060. Definition 2.1. Simple Interest: Interest earned on the original investment amount only, defined as I = P rt, I = the interest earned, P = the amount invested (principal), r = the interest rate (in dec. form), and t = the time of investment. Compound Interest: Interest earned on both the original investment amount plus previously added interest: A = P (1 + r mt , m) where A = the accumulated amt. (FV), P = the principal (PV), r = the interest rate (in dec. form), m = the number of times the account is compounded per year and t = the number of years of investment. Example 2.3. If $4,000 is invested in an account paying 2.0% compounded monthly, how much will be in the account at the end of 10 years? Continuously Compounded Interest A = P ert where P is principal, r is the annual interest rate compounded continuoulsy (as a decimal), t is time, and A is the accumulated amount at the end of t years. Example 2.4. What amount will an account have after five years if $5000 is invested at an annual rate of 2.25% compounded continuously? Effective Yield 3 Suppose a principal P earns interest at the annual rate of r, expressed as a decimal, and interest is compounded m times a year. Then the amount F after 1 years is P (1 + r/m)m ; This is the same amount that is obtainable if the same principal P is invested for one year at an annual rate of ref f = (1+r/m)m −1. ref f is called the effective annual yield. The r is often referred to as the nominal rate. Example 2.5. If $1000 is invested at an annual rate of 9% compounded monthly, then at the end of a year there is 12 0.09 = $1093.81 F = $1000 1 + 12 in the account. This is the same amount that is obtainable if the same principal of $1000 is invested for one year at an annual rate of 9.381%. We call the rate 9.381% the effective annual yield. The 9% annual rate is often refered to as the nominal rate. TVM SOLVER We can use the TVM Solver on our calculator to solve problems involving compound interest. (We cannot use TVM Solver for simple interest calculations, only compound interest.) Compound Interest with TVM Solver on the Calculator: • Press APPS. • Select Finance... (Option 1) • Select TVM Solver (Option 1) N = total number of compoundings during the time period I% = interest rate (as a percent) PV = initial amount in the account PMT = regularly occuring payments F V = future value of the account P/Y = C/Y = the number of payments/compoundings per year END/BEGIN (Note: END should be higlighted) • Enter the known information. • Scroll to the line representing the unknown data • Press SOLVE (ALPHA ENTER) Remark 2.1. Important Note: In the TVM Solver, the values for PV, PMT, and FV will sometimes be negative. This is done to represent the transfer or flow of money. We will usually look at these problems from the standpoint of the investor or borrower. e.g. a negative number represents an of money away from the investor or borrower, i.e., when money is leaving your pocket. Use a negative number when: Making payments Depositing money in a bank 4 Example 2.6. Suppose $29,000 is deposited into an account paying 7.5% annual interest. (a) How much will be in the account after 5 years if the account earns simple interest? (b) How much will be in the account after 5 years if the account is compounded monthly? (c) How much more interest is earned when the account is compounded monthly? Example 2.7. Suppose $29,000 is deposited into an account paying 7.5% annual interest. (a) How much will be in the account after 5 years if the account is compounded continuously? (b) How much interest is earned on the account? 3 Logarithms Definition 3.1. A function f is said to be a one-to-one function if each range value corresponds exactly to one domain value. Remark 3.1. Horizontal Line Test: If every horizontal line intersects the graph of a function f in no more than one place, then f is a one-to-one function. Definition 3.2. If f is a one-to-one function, then the inverse of f is the function formed by interchanging the independent and dependent variables for f. Thus, if (a, b) is a point on the graph of f, then (b, a) is a point on the graph of the inverse of f. 5 Remark 3.2. NOTE: If f is one-to-one, then the graph of its inverse, g, is a reection about the line y = x. If f is not one-to-one, then f has no inverse. Definition 3.3. Let b be a positive number with b 6= 1. If x > 0, the logarithm base b of x, denoted by logb x, is dened as y = logb x ⇔ x = by . Example 3.1. Graph f (x) = 10x and its inverse on the same graph. Definition 3.4. Common Logarithm LOG : b = 10 ⇒ y = log10 x = log x ⇔ 10y = x; Natural Logarithm LN : b = e ⇒ y = loge x = lnx ⇔ ey = x ; Change of Base Formula: logb a = log b log a = ln b ln a ; Example 3.2. Change each exponential equation to an equivalent logarithmic form: (a) 3x = 9, (b) 10x = 0.001. Example 3.3. : Solve for x, y, or b without a calculator: (a) log x = 3, (b) ln x = 0, (c) logb e2 = 2, 6 (d) log3 81 = y. Proposition 3.1. Properties of Logarithmic Functions: Let b > 0, b 6= 1, where m and n are positive real numbers. Then, logb (mn) = logb ( m n)= logb (mn ) = logb 1 = logb b = blogb x = if x > 0, x logb b = logb m = logb n if and only if m = n. Example 3.4. Write 12 (ln 2 + ln x − ln y) as a single logarithm. Example 3.5. Given logb 2 = 1.346 and logb 5 = 3.876, nd the following: (a) logb 50; (b) logb ( b42 ). Example 3.6. How long will it take for the amount in an account to double if the money is compounded continuously at an annual interest rate of 5.2%? 7