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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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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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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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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Solving Exponential and Logarithmic Equations Exponential Equations (variable in exponent position) 1. Isolate the exponential portion ( base exp onent ): Move all non-exponential factors or terms to the other side of the equation. 2. Take ln or log of each side of the equation. • Make sure to use ln if the base is “e”. Then remember that ln e = 1 . • Make sure to use log if the base is 10. • If the base is neither “e” nor “10”‚ use either ln or log‚ your choice.. 3. Bring the power (exponent)
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What factors helped to promote America’s huge industrial growth during the period from 1860-1900? America had a huge industrial revolution in the late 1800"s. Many changes happened to our great nation‚ which factored into this. The evidence clearly shows that advancements in new technology‚ a large wave of immigrants into our country and new views of our government‚ helped to promote America’s huge industrial growth from the period of 1860-1900. Advancements in new technology clearly
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create economic growth. However its actual effectiveness at meeting this objective is arguably not that good for a number of reasons which will be discussed in this essay. The main part of fiscal policy in order to increase growth is expansionary fiscal policy. This is where the government is spending more or cutting taxes in order to put more money into the economy than it is taking out. This is normally pursued when a government is trying to combat recession and increase economic growth. This should
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Q1. Sales of a particular product (in thousands of dollars) for the years 2009 through 2012 havebeen $48‚000‚ $64‚000‚$67‚00 and $83‚000‚ respectively a) What sales would you predict for 2013‚ using a simple four-year moving average? F2013 = = $65‚500 $65‚000 is the forecast for 2013 b) What sales would you predict for 2013‚ using a weighted moving average with weights of0.50 for the immediate preceding year and 0.3‚ 0.15‚ and 0.05 for the three years before that? F2013 = 0
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