Exponential Functions
Exponential functions are mathematical functions of the form f(x) = a^x, where the base "a" is a positive constant. These functions grow or decay at an increasingly rapid rate as x increases or decreases. They are characterized by their unique property of having a constant ratio between consecutive values, making them useful for modeling phenomena with exponential growth or decay.
7 Key excerpts on "Exponential Functions"
- Craig Adam(Author)
- 2011(Publication Date)
- Wiley(Publisher)
...Before discussing some of these applications, the nature and meaning of these functions must be studied in some detail and the rules for their algebraic manipulation established. 3.1 Origin and definition of the exponential function The exponential and logarithmic (log) functions are based on the idea of representing a number as the power of another number called the base. The power itself is also sometimes called the index or exponent. So we can write the number, y, in terms of x and a base number, A, as: What number should we choose as the base and why should we wish to generate such functions anyway? The answer to the second part is that these functions turn out to have interesting properties, which we can apply and use to model real life situations. Our choice of base depends on what we wish to use these functions for. For example, let us set A = 10, as we work in a decimal system. This means that y (0) = 10 0 = 1, y (1) = 10 1 = 10 etc. We can evaluate this function for all real numbers, whether they are positive or negative. A graph of this function may be plotted, as shown in Figure 3.1. This reveals the characteristic “exponential” behaviour of a function that increases very rapidly, and increasingly so, as x gets larger. On the other hand, with a negative exponent the curve shows characteristic “decay” behaviour. Note that the exponential function of a real number never has a negative value. Figure 3.1 Graphs of functions y (x) = A x By choosing numbers other than 10 it turns out that we can produce similar functions. However, there is one special base that has the additional and very useful property that its rate of change – the tangential gradient at any point on the graph of the function – is equal to the value of the function at that point...
eBook - ePub- R. Stafford Johnson(Author)
- 2013(Publication Date)
- Bloomberg Press(Publisher)
...Appendix B Uses of Exponents and Logarithms Exponential Functions An exponential function is one whose independent variable is an exponent. For example: where: y = dependent variable t = independent variable b = base (b > 1) In calculus, many Exponential Functions use as their base the irrational number 2.71828, denoted by the symbol e: An exponential function that uses e as its base is defined as a natural exponential function. For example: These functions also can be expressed as: In calculus, natural Exponential Functions have the useful property of being their own derivative. In addition to this mathematical property, e also has a finance meaning. Specifically, e is equal to the future value (FV) of $1 compounded continuously for one period at a nominal interest rate (R) of 100 percent. To see e as a future value, consider the future value of an investment of A dollars invested at an annual nominal rate of R for t years, and compounded m times per year. That is: (B.1) If we let A = $1, t = one year, and R = 100 percent, then the FV would be: (B.2) If the investment is compounded one time (m = 1), then the value of the $1 at end of the year will be $2; if it is compounded twice (m = 2), the end-of-year value will be $2.25; if it is compounded 100 times (m = 100), then the value will be 2.7048138. As m becomes large, the FV approaches the value of $2.71828. Thus, in the limit: (B.3) If A dollars are invested instead of $1, and the investment is made for t years instead of one year, then given a 100 percent interest rate the future value after t years would be: (B.4) Finally, if the nominal interest rate is different than 100 percent, then the FV is: (B.5) To prove Equation (B.5), rewrite Equation (B.1) as follows: (B.6) If we invert R/m in the inner term, we get: (B.7) The inner term takes the same form as Equation (B.2). As shown earlier, this term, in turn, approaches e as m approaches infinity...
eBook - ePubDifferentiating Instruction in Algebra 1
Ready-to-Use Activities for All Students (Grades 7-10)
- Kelli Jurek(Author)
- 2021(Publication Date)
- Routledge(Publisher)
...expressions, ➤ that Exponential Functions have a variable as the exponent, ➤ that Exponential Functions have growth and decay factors or multipliers, ➤ that exponential growth functions approach zero as x -values decrease, and ➤ that exponential decay functions approach zero as x -values increase. As a result of this unit, students will understand that: ➤ many real-life situations are modeled in Exponential Functions, and ➤ for an equation in the form of y = a (b) x, a represents the starting value, the value of b reflects either exponential decay or growth, and x is the time period. As a result of this unit, students will be able to: ➤ solve exponent functions and simplify expressions by applying the exponent rules; ➤ recognize and describe relationships in which variables grow and decay exponentially; ➤ describe how the values of a and b affect the graph of an equation in the form of y = a (b) x ; ➤ recognize exponential relationships in tables, graphs, and equations; and ➤. determine the growth and decay rates in exponential situations. Launch Scenarios ➤ After Erin graduated from college, she bought a house for $210,000. If it is estimated that real estate is appreciating in value by 5% per year, how much will the house be worth in 10 years when she plans to sell it? (Lesson 2) ➤ When Kevin's daughter is born, he and his wife invest $10,000 in an interest-bearing account for their daughter's education. If the account is earning 4% interest compounded quarterly, and no additional money is deposited into the account, how long will it take for the account to have $15,000? (Lesson 2) ➤ Katie bought a new car for $30,000. If its value is decreasing by 8% each year, how much will the truck be worth after 3 years? (Lesson 3) ➤ It is estimated that the human body can reduce caffeine in the bloodstream at a rate of 15% per hour...
eBook - ePub- Arsen Melkumian(Author)
- 2012(Publication Date)
- Routledge(Publisher)
...3 Exponential and logarithmic functions Logarithmic functions are indispensable in economic analysis as they can transform multiplicative relationships between economic variables into additive ones. In addition, economists often choose (for the sake of convenience) to optimize the natural log of an objective function instead of the objective function itself. Exponential Functions are very useful when modeling the growth of a certain economic variable. For example, we can use Exponential Functions to model the growth of the population of a country. This chapter introduces logarithmic and Exponential Functions and closes with some Mathematica examples. 3.1 Logarithmic function Consider the following equation: In the equation above the base is equal to 3, and the exponent is equal to 5. The power to which 3 must be raised to yield 243 is called the logarithm (or log) to the base 3 of 243. So, logarithm to the base 3 of 243 is equal to 5: In general, if where a, B > 0 and a ≠ 1, and x ∈ ℝ, then we can write that For example, 2 5 = 32 implies that log to the base 2 of 32 equals 5 E XAMPLE 3.1 (a) log 2 1 = 0, since 2 raised to the power of 0 is 1. (b) log 7 7 = 1, since 7 raised to the power of 1 is 7. (c) log 10 = −1, since 10 raised to the power of −1 is. (d) log 3 81 = 4, since 3 raised to the power of 4 is 81. (e) log 5 (−25) is not defined, since −25 < 0. (f) log 12 1728 = 3, since 12 raised to the power of 3 is 1728. (g) log 5 = −2, since 5 needs to be raised to the power of −2 in order to get. Often economists work with logarithms to the base e, where e is the irrational number 2.718 … known as the exponential. Logarithms to the base e are referred to as natural logarithms. We can write either log e B or ln B to refer to the natural logarithm of B. Now, consider the function y = f (x) = ln x. The graph of y is given in Figure 3.1...
eBook - ePub- Alan J. Cann(Author)
- 2013(Publication Date)
- Wiley(Publisher)
...To calculate the value of 10 0.65 type this expression into your calculator and you will see that it equals 4.47. However, the reason for the explanations in this chapter is that, even with a calculator, you still need to understand how exponents work. So what has all this got to do with the real world? Here is a practical example you may be interested in. If you invested £1200 in a bank account which pays 6% interest compounded annually, how much money would be in your account after 5 years? If the annual interest rate is 6 per cent, this can be written as 0.06. If you started with £1, at the end of one year, you would have 1 + 0.06 = £1.06. At the end of the second year, you would have £1.06 + 0.06 * (1 + 0.06) = (1.06) * (1.06) = (1.06) 2. At the end of 5 years, you would have (1.06) 5 = £1.33 for each pound invested, so: 6.2. Exponential Functions The expression y = z x solved for y in terms of x does not graph as a straight line on normal graph paper (Figure 6.1). Any quantity which increases by being multiplied by the same value at regular intervals is said to grow ‘exponentially’, i.e. each subsequent value is equal to the previous value multiplied by a constant, z : Exponential Functions occur frequently in biology because they describe processes of growth and decay, for example in radioactive decay or bacterial growth. Figure 6.1 An exponential graph Radioactive decay Radioisotopes are unstable forms of elements, atoms of which decay spontaneously at random. Radioactive decay is described by the function: where N is the amount of radioactivity remaining at time t, N 0 the original amount of radioactivity and λ the decay constant for the particular radionuclide. Since the speed at which the element decays is constant, this is an exponential function. The term ‘half-life’ is used to describe the time taken for 50 per cent of the atoms in a sample to decay...
eBook - ePub- John Bird(Author)
- 2019(Publication Date)
- Routledge(Publisher)
...Chapter 9 Logarithms and Exponential Functions Why it is important to understand: Logarithms and Exponential Functions All types of engineers use natural and common logarithms. Chemical engineers use them to measure radioactive decay and pH solutions, both of which are measured on a logarithmic scale. The Richter scale which measures earthquake intensity is a logarithmic scale. Biomedical engineers use logarithms to measure cell decay and growth, and also to measure light intensity for bone mineral density measurements. In electrical engineering, a dB (decibel) scale is very useful for expressing attenuations in radio propagation and circuit gains, and logarithms are used for implementing arithmetic operations in digital circuits. Exponential Functions are used in engineering, physics, biology and economics. There are many quantities that grow exponentially; some examples are population, compound interest and charge in a capacitor. With exponential growth, the rate of growth increases as time increases. We also have exponential decay; some examples are radioactive decay, atmospheric pressure, Newton’s law of cooling and linear expansion. Understanding and using logarithms and Exponential Functions is therefore important in many branches of science and engineering. At the end of this chapter, you should be able to: define base, power, exponent, index and logarithm distinguish between common and Napierian (i.e. hyperbolic or natural) logarithms state the laws of logarithms simplify logarithmic expressions solve equations involving logarithms solve indicial equations sketch graphs of log 10 x and log e x evaluate Exponential Functions using a calculator plot graphs of Exponential Functions evaluate Napierian logarithms using a calculator solve equations involving Napierian logarithms appreciate the many examples of laws of growth and decay in engineering and science perform calculations involving the laws of growth and decay Science and Mathematics for Engineering...
eBook - ePub- Kenneth W. Wachter(Author)
- 2014(Publication Date)
- Harvard University Press(Publisher)
...A demographer who says that the world’s growth rate peaked at “22” means it peaked at “22 per thousand per year”. 1.3 The Exponential Curve We return now from our discussion of growth rates to our formula for the population over time in the case when the ratios of births and deaths to population remain constant. We rewrite this formula in the form Here Rt is the “exponent” of the number “e”. We are raising the number “e” to the power Rt, and another notation for this quantity is exp(Rt), the “exponential function” of Rt. The exponential function is the inverse function for natural logarithms. That is to say, for any x or y, We know from our linear equation that log(A) is R, so we have These facts about logarithms show us that our equation with A and our equation with e are the same equation. The graph of exp(Rt) as a function of t is called the “exponential curve”, the continuous-time version of the curve for geometric growth. It is important to become familiar with its shape. The shape depends on whether R is greater than, equal to, or less than zero, as shown in Figure 1.3. If the ratios B/K and D/K remain unchanged, the population either grows faster and faster beyond all bounds, stays the same, or heads toward zero steeply at first and more gradually later. The most common case is the case in the top panel when births exceed deaths and R >0. This is the case of exponential growth that leads many people to feel alarm at the future prospects for humanity on the planet. Our formula for exponential growth treats the growth rate R as a constant. To apply it in practice, we break up any long period of time into shorter intervals within which the growth rate does stay more or less constant and proceed step by step. The graph of the logarithm of population size becomes a series of straight-line pieces. In each step we “run” forward in time by some amount n, and our graph has a corresponding “rise” equal to n times R...
