LOGARITHMIC AND EXPONENTIAL FUNCTIONS Inverse relations Exponential functions Exponential and logarithmic equations One logarithm THE LOGARITHMIC FUNCTION WITH BASE b is the function y = logb x. b is normally a number greater than 1 (although it need only be greater than 0 and not equal to 1). The function is defined for all x > 0. Here is its graph for any base b. Note the following: • For any base‚ the x-intercept is 1. Why? To see the answer‚ pass your mouse over the colored
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Adjusted Present Value Adjusted present value is an investment appraisal technique similar to net present value method. However‚ instead of using weighted average cost of capital as the discount rate‚ ungeared cost of equity is used to discount the cash flows from a project and there is an adjustment for the tax shield provided by related debt capital. Formula Adjusted Present Value = PV of Cash Flows using Ungeared Cost of Equity + Present Value of Tax Shield Where PV stands for ’present value’
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EXPONENTIAL AND LOGARITHMIC FUNCTIONS I.EXPONENTIAL FUNCTION A. Definition An exponential function is a function defined by f(x) = ax ‚ where a > 0 and a ≠ 1. The domain of the function is the set of real numbers and the range is the set of positive numbers. B. Evaluating Exponential Functions 1. Given: f(x) = 2x‚ find a. f(3) = ____ b. f(5) = _____ c. f(-2) = ______ d. f(-4) = ______ 2. Evaluate f(x) = ( 1)x if 2 a. x = 2 ____ b. x = 4 _____ c. x = -3 ______ d
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Exponential and Logarithmic Functions * Verify that the natural logarithm function defined as an integral has the same properties as the natural logarithm function earlier defined as the inverse of the natural exponential function. Integrals of Exponential and Logarithmic Functions Function | Integral | lnx | x ∙ lnx - x + c | logx | (x ∙ lnx - x) / ln(10) + c | logax | x(logax - logae) + c | ex | ex+c | ek∙x | 1 / k ∙ ek∙x + c | ax | ax / lna + c | xn | 1 / (n+1) ∙ xn+1 +
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21. Optimization 2 Test 3 22. Exponential Functions 23. Logarithmic Functions 24. Compound Interest 25. Differentiation of Exponential Functions 26. Differential of Logarithmic Functions 27. Exponential Functions as Mathematical Models
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I O N of RISK-ADJUSTED DISCOUNT RATES and LIABILITY BETA RUSSELL E. BINGHAM T H E H A R T F O R D FINANCIAL SERVICES G R O U P Table of Contents Page 2 3 5 7 8 11 12 13 14 14 15 16 17 17 18 Subject Abstract 1. Summary 2. Total Return Model 3. After-Tax Discounting 4. Derivation of Risk-Adjusted Discount Rate and Liability Beta Figure l : Baseline Risk / Return Line vs Leverage 5. Liability Beta Figure 2: Equity vs Liability Beta Figure 3: Equity Beta vs Risk-Adjusted Discount Rate (After-Tax)
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MATH133 Unit 5: Exponential and Logarithmic Functions Individual Project Assignment: Version 2A Show all of your work details for these calculations. Please review this Web site to see how to type mathematics using the keyboard symbols. IMPORTANT: See Question 1 in Problem 2 below for special IP instructions. This is mandatory. Problem 1: Photic Zone Light entering water in a pond‚ lake‚ sea‚ or ocean will be absorbed or scattered by the particles in the water and its intensity‚ I‚ will be attenuated
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Exponential Functions in Business Turgenbayeva Aiida ID 20092726 Variant 2 Kazakhstan Institute of Management‚ Economics and Strategic Research MSC1101 Mathematics for Business and Economics Instructor: Dilyara Nartova Section #2 Summer-I 2009 Abstract This project reflects my knowledge and understanding of the interest rate‚ its types‚ formula and its evaluation in order to determine the most profitable type of investment scheme for National Bank wishing to increase
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Name:________________________________ Part 1 Exponential Functions Project There are three parts to this project. You must complete Part 1 (60 points)‚ but you may choose to do either Part 2 or Part 3 (40 points each). You may also do all three parts for a total of 140 points; however‚ you must fully complete either Part 2 or Part 3 to get credit (NOT ½ of Part 2 and ½ of Part 3). This project is due on December 5th. Turning it in late forfeits your right to extra credit and there will be
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lay in developing strategic relationships with the aristocracy‚ the towns‚ and the church. The worldwide trend of urbanization has changed over time as well as changing the functions of cities. The cities represent a world of opportunity which links with urbanization and economic growth. The cities are also home to a high concentration of povertys. The urban areas have
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