Exponential Rules
Exponential rules are a set of principles that govern the manipulation and simplification of expressions involving exponents. These rules include properties such as the product rule, power rule, and quotient rule, which allow for the efficient computation of exponential expressions. Understanding and applying these rules is essential for solving equations and working with exponential functions.
5 Key excerpts on "Exponential Rules"
eBook - ePubDifferentiating Instruction in Algebra 1
Ready-to-Use Activities for All Students (Grades 7-10)
- Kelli Jurek(Author)
- 2021(Publication Date)
- Routledge(Publisher)
...Unit 3 Exponent Rules and Exponential Functions DOI: 10.4324/9781003234180-4 In this unit, students will continue their study of functions by identifying exponential growth and decay situations in tables, graphs, and equations. Real-life applications introduced through story problems are included in most lessons. The final project, which could be used as an authentic assessment, is a children's book that students will write to demonstrate their understanding of exponential functions. This unit begins with a preassessment and four real-life applications of exponential functions that can be discussed in small groups and then as a larger group. Many of the activities will offer the students an opportunity to choose learning activities according to their learning style, personal interests, and readiness level. What Do We Want Students to Know? Common Core State Standards Addressed: • 8.EE.1 • A.CED.1 • A.SSE.3c • F.IF.7e, 8b • F.LE.1a, 1c, 2, 5 Big Ideas • There are exponent rules that allow us to solve complicated-looking expressions. • Exponential functions are different from linear equations. • Exponential functions always have a variable in the exponent. • The base in the equation determines whether the function will grow or decay. • There are many real-life situations that can be modeled with exponential functions. Essential Questions • What are the exponent rules and how do I use them? • How are linear and exponential functions the same and different? • How do the base and the exponent affect the shape of the graph? • In real-world situations, how do I know what information goes into the equation and where does it go? Critical Vocabulary Base Exponential decay Standard form Exponent Compound growth Initial amount Exponential growth Growth factor Decay factor Unit Objectives As a result of this unit, students will know: ➤ how to apply the exponent rules to simplify...
- 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- Alan J. Cann(Author)
- 2013(Publication Date)
- Wiley(Publisher)
...6 Exponents and Logs L EARNING O BJECTIVES : On completing this chapter, you should be able to: understand and be able to manipulate exponents; understand and be able to manipulate logarithms; be able to use logarithms to perform calculations; 6.1. Exponents An exponent (also called the ‘power’ or ‘index’) of a number indicates how many times a number or term (the ‘base’) should be multiplied by itself. Just as multiplication is a shortcut for addition: So exponents are a shortcut for multiplication: Similarly, and and As you will see later, logarithms are a shortcut for exponents, since the log function is the inverse of the exponential. How to manipulate and use exponents can be summarized in three rules. 1. Rule 1 – to multiply identical bases, add the exponents, e.g. 2. Rule 2 – to divide identical bases, subtract the exponents, e.g. 3. Rule 3 – when there are two or more exponents and only one base, multiply the exponents, e.g. Remember, these rules apply to identical bases only – you cannot apply them to different bases. Note that zero raised to any power always equals zero (0 n = 0) and that any number raised to the power zero equals one (n 0 = 1). Negative exponents are the inverse of numbers raised to a positive integer: Whenever you see a negative exponent, this should immediately suggest that the expression has a value of less than 1, e.g. Working in powers of 10: Fractional exponents can be dealt with in exactly the same way as integer exponents: The square root of any number = n 0.5, e.g. and the cube root of any number = n 1/3, e.g. Using the rules of exponents: In real life no one expects you to work out complex exponents by hand – use a scientific calculator. 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...
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 - 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...
