5.1: Prelude to Exponential and Logarithmic Functions - Mathematics

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;, 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.

  1. 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 .


  • Attendance and class participation 5%
  • Online Assignments 5%
  • Quizzes 10%
  • Midterm 1 exam 15%
  • Midterm 2 exam 15%
  • Final exam 50%


Students should be aware that they have certain rights to confidentiality concerning the return of course papers and the posting of marks.
Please pay careful attention to the options discussed in class at the beginning of the semester.


I-Clicker+ (Available at the SFU Bookstore) ISBN: 97814641201

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.

Download our free learning tools apps and test prep books

Names of standardized tests are owned by the trademark holders and are not affiliated with Varsity Tutors LLC.

4.9/5.0 Satisfaction Rating over the last 100,000 sessions. As of 4/27/18.

Media outlet trademarks are owned by the respective media outlets and are not affiliated with Varsity Tutors.

Award-Winning claim based on CBS Local and Houston Press awards.

Varsity Tutors does not have affiliation with universities mentioned on its website.

Varsity Tutors connects learners with experts. Instructors are independent contractors who tailor their services to each client, using their own style, methods and materials.

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

WHY. We are a small, independent publisher founded by a math teacher and his wife. We believe in the value we bring to teachers and schools, and we want to keep doing it. We keep our prices low so all teachers and schools can benefit from our products and services. We ask that you help us in our mission by complying with these Terms & Conditions.

PLEASE, NO SHARING. We know it’s nice to share, but please don’t share your membership content or your login or validation info. Your membership is a Single User License, which means it gives one person – you -- the right to access the membership content (Answer Keys, editable lesson files, pdfs, etc.) but is not meant to be shared.

  • Please do not copy or share the Answer Keys or other membership content.
  • Please do not post the Answer Keys or other membership content on a website for others to view. This includes school websites and teacher pages on school websites.
  • You can make copies of the Answer Keys to hand out to your class, but please collect them when the students are finished with them.
  • If you are a school, please purchase a license for each teacher/user.

PLEASE RESPECT OUR COPYRIGHT AND TRADE SECRETS. We own the copyright in all the materials we create, and we license certain copyrights in software we use to run our site, manage credentials and create our materials some of this copyrighted software may be embedded in the materials you download. When you subscribe, we give you permission (a “Single User License”) to use our copyrights and trade secrets and those we license from others, according to our Terms & Conditions. So in addition to agreeing not to copy or share, we ask you:

  • Please don’t reverse-engineer the software and please don’t change or delete any authorship, version, property or other metadata.
  • Please don’t try to hack our validation system, or ask anyone else to try to get around it.
  • Please don’t put the software, your login information or any of our materials on a network where people other than you can access it
  • Please don’t copy or modify the software or membership content in any way unless you have purchased editable files
  • If you create a modified assignment using a purchased editable file, please credit us as follows on all assignment and answer key pages:

“This assignment is a teacher-modified version of [eMath Title] Copyright © 201x eMATHinstruction, LLC, used by permission”

FEEDBACK REQUESTED. We value your feedback about our products and services. We think others will value it, too. That’s why we may do the following (and we ask that you agree):

  • Use your feedback to make improvements to our products and services and even launch new products and services, with the understanding that you will not be paid or own any part of the new or improved products and services (unless we otherwise agree in writing ahead of time).
  • Share your feedback, including testimonials, on our website or other advertising and promotional materials, with the understanding that you will not be paid or own any part of the advertising or promotional materials (unless we otherwise agree in writing ahead of time).

SATISFACTION GUARANTEED. If you are not 100% satisfied, we will refund you the purchase price you paid within 30 days. To get a refund:

  • Within 30 days of your purchase,
  • Delete the software and all membership content from all your computers, destroy all photocopies or printouts of our materials and return all tangible copies (disks, workbooks, etc) and other materials you have received from us to:

eMATHinstruction Returns Department
10 Fruit Bud Lane
Red Hook, NY 12571

TECHNICAL SUPPORT: If you are having trouble logging in or accessing your materials, or if your downloaded materials won’t open or are illegible, please notify us immediately by email at [email protected] so we can get it fixed.

NO WARRANTY. We believe in the quality and value of our products and services, and we work hard to make sure they work well and are free of bugs. But that said, we are providing our products and services to you “as is,” which means we are not responsible if something bad happens to you or your computer system as a result of using our products and services. For our full Disclaimer of Warranties, please see our Legalese version of these Terms & Conditions Here .

DISPUTES. If we have a dispute that we cannot resolve on our own, we will use Binding Arbitration instead of filing a lawsuit in a regular court (except that you can use small claims court). Binding Arbitration means our case will be decided by one or more arbitrators who are chosen and paid by all parties to the dispute. Arbitration is a faster and less formal way of resolving disputes and therefore tends to cost less.

  • To begin an arbitration proceeding, please send a letter requesting arbitration and describing your claim to:

Emath Instruction Inc.
10 Fruit Bud Lane
Red Hook, NY 12571

LIMITATION OF LIABILITY. If you do win a case against us, the most you can recover from us is the amount you have paid us.

To see the Legalese version of our Terms & Conditions, please click HERE. We’ve given you the highlights above, in plain English, but it’s a good idea to look at the Legalese, too, because by checking the box below and proceeding with your purchase you are agreeing to both the English and Legalese.

Thank you for using eMATHinstruction materials. In order to continue to provide high quality mathematics resources to you and your students we respectfully request that you do not post this or any of our files on any website. Doing so is a violation of copyright.

The content you are trying to access requires a membership. If you already have a plan, please login. If you need to purchase a membership we offer yearly memberships for tutors and teachers and special bulk discounts for schools.

Sorry, the content you are trying to access requires verification that you are a mathematics teacher. Please click the link below to submit your verification request.

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.

NumberExponential ExpressionLogarithm
100010 3 3
10010 2 2
1010 1 1
110 0 0
1/10 = 0.110 - 1 -1
1/100 = 0.0110 - 2 -2
1/1000 = 0.00110 - 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)

  1. enter the number,
  2. press the inverse (inv) or shift button, then
  3. 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?


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?

Watch the video: Concepts of Exponential u0026 Logarithmic Fn. CBSE 12 Maths u0026comp. Ex intro (December 2021).