In this chapter we examine exponential and logarithmic functions. We will need these functions in the next chapter, when examining financial calculations.
This chapter is a new addition to this textbook. The California Community Colleges Curriculum Course Descriptor for Finite Mathematics (C-ID; https://c-id.net/descriptors.html, http://www.ccccurriculum.net/articulation/) now requires coverage of exponential and logarithmic functions in a Finite Mathematics course that is part of an Associate Degree for Transfer.
Students enrolling in Finite Mathematics typically are required to complete an Intermediate Algebra course or equivalent, as a prerequisite, so students have already been exposed to much of the material in this chapter. However many students require a review of this material, which is the basis for financial calculations based on compound interest in the following chapter. In addition, review of this material is particularly important at colleges where Finite Mathematics serves as a prerequisite for Business Calculus.
This book assumes students have mastered working with exponents, and properties of exponents; it focuses on review of exponential and logarithmic functions with an eye toward skills needed to use exponential growth and decay models for financial calculations and other business applications, as well as subsequent use in a course on Business Calculus. For the most part, financial applications are not stressed in this new chapter, as financial calculations are the focus of the following chapter.
How to do exponential and logarithmic curve fitting in Python? I found only polynomial fitting
I have a set of data and I want to compare which line describes it best (polynomials of different orders, exponential or logarithmic).
I use Python and Numpy and for polynomial fitting there is a function polyfit() . But I found no such functions for exponential and logarithmic fitting.
Are there any? Or how to solve it otherwise?
5.2 The Exponential and Logarithm Functions
Definition 2: The exponential function is defined for every complex number z to be the value of the series .
For example, this means that exp(0) = 1.
Proposition 6: For all complex numbers and , one has
Proof: By Proposition 5, we know that is absolutely convergent and converges to . On the other hand, the inside sum is by the binomial theorem. So, the series is which completes the proof.
Corollary 1: The exponential function is continuous at every complex number.
Proof: By the addition theorem, one has for every complex number a:
In particular, if the exponential function is continuous at 0, then it will be continuous everywhere. Let . Choose so that and . Then for all z with , one has
So, the exponential function is indeed continuous at 0.
Corollary 2: As a function of real numbers, the exponential function is an increasing function, i.e. exp(x) 0. But this is clear from the series since it starts out as 1 + (y - x) and all the remaining terms are positive.
Corollary 3: Again restricting to real values, the range of the exponential function is the set of all positive real numbers. Clearly, the exponential function can be made arbitrarily large by taking x sufficiently large and positive. Since exp(-x) = 1/exp(x) by the addition theorem, it follows that one can make the exponential function arbitrarily small by taking x negative and large in absolute value. Since the exponential function is continuous, the result now follows from the intermediate value theorem.
It follows from Corollary 3 that the exponential function has an inverse function defined for all positive real numbers and with range the set of all real numbers. This function is denoted is called the logarithm function.
Definition 3: Let a be a positive real number. Then define the power function for real numbers x.
- is an increasing function.
Proof: These are simple consequences of the definition of the logarithm function as the inverse of the exponential theorem. For example,
Applying the logarithm to both sides gives the second assertion.
Similarly, shows the third assertion.
For the last assertion, suppose that . Then, since the exponential function is increasing, we have . But then, using the fact that the exponential function is the inverse of the logarithm function, we get . Taking the contrapositive, we see that we have shown that if , then .
Assertion ii is proved similarly.
iii. One has . So,
iv. One has .
Proposition 7: The natural logarithm function is continuous at all x > 0.
Proof: The hard part of the proof is to show that is continuous at x = 1. So, let's assume we have done that part. Let and let us then show that is continuous at . For this, let and, using the continuity at 1, we know that there is a such that provided that . Let . Suppose that x is any positive real number such that . Then and so . Since the left side of the inequality is just , we see that is continuous at .
- Suppose x
Solving the inequality, we get and so . Since the natural logarithm function is increasing, it follows that and so
because . Solving, we get and so applying the logarithm function to both sides, we get
as desired. This completes the proof of the proposition.
Proposition 8: The Euler constant e can be represented as a limit:
Proof: Using the binomial theorem, one has the identity
So, the basic idea is to show that the additional products in each term do not matter. To see this, choose and start with
Lemma 1: If for are non-negative real numbers, then .
Proof: Clearly, the result is true for m = 1. If it is not true for arbitrary positive integers m, then there would be a smallest positive number m for which it were false. Since m > 1, it would be true for m - 1. So
Applying Lemma 1, one can estimate the product as:
where we have used the formula for the sum of an arithmetic series and where the last inequality holds for all n with . In particular, it holds when . One has:
where T is the largest integer less than or equal to and where we have chosen n large enough so that .
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Logarithmic to Exponential Form
Logarithmic functions are inverses of exponential functions . So, a log is an exponent !
y = log b x if and only if b y = x for all x > 0 and 0 < b &ne 1 .
Write log 5 125 = 3 in exponential form.
Write log z w = t in exponential form.
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Sample Spaces and Probability
If two coins are tossed, what is the probability that both coins will fall heads? The problem seems simple enough, but it is not uncommon to hear the incorrect answer of . A student may incorrectly reason that if two coins are tossed there are three possibilities, one head, two heads, or no heads. Therefore, the probability of two heads is one out of three. The answer is wrong because if we toss two coins there are four possibilities and not three. For clarity, assume that one coin is a penny and the other a nickel. Then we have the following four possibilities:
The possibility HT, for example, indicates a head on the penny and a tail on the nickel, while TH represents a tail on the penny and a head on the nickel.
It is for this reason, we emphasize the need for understanding sample spaces.
An act of flipping coins, rolling dice, drawing cards, or surveying people are referred to as an experiment.
Algebra Formulas for Class 10
An important algebra formula introduced in class 10 is the &ldquo quadratic formula &rdquo. The general form of the quadratic equation is ax 2 + bx + c = 0, and there are two methods of solving this quadratic equation. The first method is to solve the quadratic equation by the algebraic method, and the second method is to solve through the use of the quadratic formula. The below formula is helpful to quickly find the values of the variable x with the least number of steps.
In the above expression, the value b 2 - 4ac is called the determinant and is useful to find the nature of the roots of the given equation. Based on the value of the determinant, the three types of roots are given below.
- If b 2 - 4ac > 0, then the quadratic equation has two distinct real roots.
- If b 2 - 4ac = 0, then the quadratic equation has two equal real roots.
- If b 2 - 4ac < 0, then the quadratic equation has two imaginary roots.
Apart from this, we have a few other formulas related to progressions . Progressions include some of the basic sequences such as arithmetic sequence and geometric sequence. The arithmetic sequence is obtained by adding a constant value to the successive terms of the series. The terms of the arithmetic sequence is a, a + d, a + 2d, a + 3d, a + 4d, . a + (n - 1)d. The geometric sequence is obtained by multiplying a constant value to the successive terms of the series. The terms of the geometric sequence are a, ar, ar 2 , ar 3 , ar 4 , . ar n-1 . The below formulas are helpful to find the nth term and the sum of the terms of the arithmetic, and geometric sequence.
Unit 6 – Exponents, Exponents, Exponents and More Exponents
This unit begins with a fundamental treatment of exponent rules and the development of negative and zero exponents. We then develop the concepts of exponential growth and decay from a fraction perspective. Finally, percent work allows us to develop growth models based on constant percent rates of change. Geometric sequences are tied to exponential growth in the last lesson.
Unit #6 Review – Exponents, Exponents, Exponents
Unit #6 Mid-Unit Quiz (Through Lesson #5) – Form A
Unit #6 Mid-Unit Quiz (Through Lesson #5) – Form B
Unit #6.Negative and Zero Exponent Practice
Unit #6 – Percent Warm-Up (Before Lesson #5)
Unit #6 – Interest Modeling Performance Task
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5.1: Prelude to Exponential and Logarithmic Functions - Mathematics
Two kinds of logarithms are often used in chemistry: common (or Briggian) logarithms and natural (or Napierian) logarithms. The power to which a base of 10 must be raised to obtain a number is called the common logarithm (log) of the number. The power to which the base e (e = 2.718281828. ) must be raised to obtain a number is called the natural logarithm (ln) of the number.
In simpler terms, my 8th grade math teacher always told me: LOGS ARE EXPONENTS!! What did she mean by that?
- Using log10 ("log to the base 10"):
log10100 = 2 is equivalent to 10 2 = 100
where 10 is the base, 2 is the logarithm (i.e., the exponent or power) and 100 is the number.
The rest of this mini-presentation will concentrate on logarithms to the base 10 (or logs). One use of logs in chemistry involves pH, where pH = -log10 of the hydrogen ion concentration.
Here are some simple examples of logs.
|1/10 = 0.1||10 - 1||-1|
|1/100 = 0.01||10 - 2||-2|
|1/1000 = 0.001||10 - 3||-3|
- Example 1: log 5.43 x 10 10 = 10.73479983. (way too many significant figures)
So, let's look at the logarithm more closely and figure out how to determine the correct number of significant figures it should have.
- Example 1: log 5.43 x 10 10 = 10.735
The number has 3 significant figures, but its log ends up with 5 significant figures, since the mantissa has 3 and the characteristic has 2.
- Example 4: What is the pH of an aqueous solution when the concentration of hydrogen ion is 5.0 x 10 - 4 M?
FINDING ANTILOGARITHMS (also called Inverse Logarithm)
- enter the number,
- press the inverse (inv) or shift button, then
- press the log (or ln) button. It might also be labeled the 10 x (or e x ) button.
- Example 5: log x = 4.203 so, x = inverse log of 4.203 = 15958.79147. (too many significant figures)
There are three significant figures in the mantissa of the log, so the number has 3 significant figures. The answer to the correct number of significant figures is 1.60 x 10 4 .
- Example 8: What is the concentration of the hydrogen ion concentration in an aqueous solution with pH = 13.22?
CALCULATIONS INVOLVING LOGARITHMS
Because logarithms are exponents, mathematical operations involving them follow the same rules as those for exponents.
Answers and Explanations to the Questions
Woodforest, Texas, a suburb of Houston, is determined to close the digital divide in its community. A few years ago, community leaders discovered that their citizens were computer illiterate. They did not have access to the internet and were shut out of the information superhighway. The leaders established the World Wide Web on Wheels, a set of mobile computer stations.
World Wide Web on Wheels has achieved its goal of only 100 computer illiterate citizens in Woodforest. Community leaders studied the monthly progress of the World Wide Web on Wheels. According to the data, the decline of computer illiterate citizens can be described by the following function:
1. How many people are computer illiterate 10 months after the inception of the World Wide Web on Wheels?
Compare this function to the original exponential growth function:
The variable y represents the number of computer illiterate people at the end of 10 months, so 100 people are still computer illiterate after the World Wide Web on Wheels began to work in the community.
2. Does this function represent exponential decay or exponential growth?
- This function represents exponential decay because a negative sign sits in front of the percent change (.12).
3. What is the monthly rate of change?
4. How many people were computer illiterate 10 months ago, at the inception of the World Wide Web on Wheels?