Exponentials and Logarithms
Exponentials and logarithms are inverse operations that are commonly used in mathematics. Exponentials involve raising a number to a certain power, while logarithms are used to find the exponent to which a specific base must be raised to produce a given number. These concepts are fundamental in various mathematical applications, including solving equations, modeling growth and decay, and analyzing data.
8 Key excerpts on "Exponentials and Logarithms"
- 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- 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- 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)
...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 - 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...
eBook - ePubUnderstanding Lesson Study for Mathematics
A Practical Guide for Improving Teaching and Learning
- Rosa Archer, Siân Morgan, David Swanson(Authors)
- 2020(Publication Date)
- Routledge(Publisher)
...finding x in 2 x = 3.44 × 10 12); the particular powers of 10; that log 10 is somehow related to the powers of 10; the connections of log 10 to standard form; and examples of multiplied numbers being equivalent to the adding of their logs. What they haven’t developed yet includes explicit awareness that logarithms are the inverse of powers, that they are one of the two distinct inverses of powers, and that log (ab) = log (a) + log (b). It would be possible for a teacher to simply tell the students these things at this point, relating the explanation to the problems they have already solved. Given their previous engagement and the understanding they have already developed they would be maximally receptive to such an intervention, and it should effectively crystallise their thinking. However, as we set out to avoid telling the students anything, what else could be done to encourage the class itself to come up with these crystallisations? With more time for the original lesson there were some opportunities to push a little further within the original tasks. For example, we could have spent a little more time looking for patterns within the logs of the first nine positive integers to create more examples related to log (ab) = log (a) + log (b), which we could then have attempted to generalise. We could also have more explicitly asked what the log button does after the log trick, as there is a possibility some students could guess this at this stage. Before doing this though, it would perhaps be better to pose a new problem: Given that log means log 10, can we find a trick for logs of a different base, i.e. log 2 ? After showing students how to use their calculator to find logs to different bases, students can play around as they have already done for log 10. This time they are highly unlikely to find a trick for doing them without a calculator (but please get in touch if you ever find one)...
- F. Xavier Malcata(Author)
- 2020(Publication Date)
- Wiley(Publisher)
...(2.15) after replacement of y by −y (since e −y is, by definition, 1/e y). A generalization of Eq. (2.15) reads (2.17) where x 1 = x 2 = ⋯ = x n = x readily implies (2.18) by virtue of the definition of multiplication as an iterated sum. Figure 2.2 Variation of (natural) (a) exponential, e x, and (b) logarithm, ln x, as a function of a real number, x. The inverse of the exponential is the logarithm of the same base, i.e. ln x for the case under scrutiny encompassing e as base; the corresponding plot is labeled as Fig. 2.2 b. A vertical asymptote, viz. (2.19) is apparent (the concept of limit will be explored in due course); the plot of ln x may be produced from that of e x in Fig. 2.2 a, via the rotational procedure referred to above. In terms of properties, one finds that (2.20) – so the logarithm converts a product to a sum; in fact, Eq. (2.20) is equivalent to (2.21) after taking exponentials of both sides, where Eq. (2.15) supports (2.22) – while the definition of inverse function, applied three times, allows one to get (2.23) as universal condition, thus guaranteeing validity of Eq. (2.20). If n factors x i are considered, then Eq. (2.20) becomes (2.24) should x 1 = x 2 = ⋯ = x n = x hold, then Eq. (2.24) simplifies to (2.25) If y is replaced by 1/ y in Eq. (2.20), then one eventually gets (2.26) – since ln { x / y } + ln y = ln { xy / y } = ln x as per Eq. (2.20), with isolation of ln { x / y } retrieving the above result; hence, a logarithm transforms a quotient into a difference. The concept of logarithm extends to bases other than e, say, (2.27) a ‐based exponentials may then be taken of both sides to get (2.28) – since a ‐based exponential and logarithm are inverse functions of each other. If b ‐based logarithms are taken of both sides, then Eq. (2.28) becomes (2.29) in agreement with Eq. (2.25) and after application to Eq. (2.28) – which may, in turn, be combined with Eq...
