# Precalculus 4: Exponentials and logarithms

## What you'll learn

- How to solve problems concerning exponentials or logarithms (illustrated with 239 solved problems) and why these methods work.
- Applications of exponential and logarithmic functions in finance, engineering, and natural sciences (some of them in my videos, some as reading material).
- Binomial Theorem with two proofs (a combinatorial one and one by induction), Pascal's Triangle, and how to apply them.
- Definitions of Euler's number e, how to approximate it, and how to prove that this number is irrational.
- Definitions and computational rules for powers with various types of exponents (natural, integer, rational, real).
- Definition of logarithms in relation to exponents, with computational rules (these will be related to the rules for powers).
- Exponential functions, their properties and graphs.
- Logarithmic functions, their properties and graphs.
- Power functions, their properties and graphs; interactions between power functions and exponential functions.
- Graph transformations for exponential and logarithmic functions, and for some power functions.
- Solving exponential equations and inequalities.
- Solving logarithmic equations and inequalities.

## Requirements

- High-school mathematics, mainly arithmetic; solving simple equations (linear and quadratic) with one unknown; some (but very few) polynomial equations will be solved during the course, but you can skip them if you want to (no problem at all; it will not affect your understanding of the main subjects of the course)
- Precalculus 1: Basic notions (mainly the concept of function and related concepts; sets; logic)
- You are always welcome with your questions. If something in the lectures is unclear, please, ask. It is best to use QA, so that all the other students can see my additional explanations about the unclear topics. Remember: you are never alone with your doubts, and it is to everybody's advantage if you ask your questions on the forum.

## Description

Precalculus 4: Exponentials and logarithms

*Mathematics from high school to university*

S1. Introduction to the course

You will learn: you will get a very general introduction to exponential and logarithmic functions: how they look and what they describe; you will learn about their simplicity in some aspects, and about some extraordinary complications you don't necessarily have to think about (but you should have a general idea about them); exponential and logarithmic functions are (next to polynomials, rational functions, power functions, trigonometric and inverse trigonometric functions you have learned about in the previous courses in the Precalculus series) examples of *elementary functions*: these functions are the building blocks for all the functions we will work with in the upcoming Calculus series.

S2.71828... The noble number *e*, the Binomial Theorem, and Pascal's Triangle

You will learn: about number *e*, both in an intuitive way (by images) and in a more formal way. In order to be able to perform some formal proofs, you will need the Binomial Theorem: theorem telling you how to raise a sum of two terms to any positive natural power; you will learn about factorial, about binomial coefficients, and Pascal's Triangle (all this will come back in the course in Discrete Mathematics, and then you will get much more practice and combinatorial problems to solve; now we just need the Binomial Theorem as a tool for dealing with *e*).

S3. Powers with various types of exponents

You will learn: about powers with natural, integer, and rational exponents and the computation rules holding for them (the product rules, the quotient rules, the power rule); you will also get an explanation of a more serious stuff: how we can be sure about existence of *n*th roots of numbers. We will gradually get more and more understanding about the topic of plotting exponential functions. You will also get plenty of exercise to get comfortable with the topic of powers with various types of exponents, and power-related computations.

S4. Power functions, their properties and graphs

You will learn: about power functions and their properties: monotonicity for positive arguments, monotonicity for negative arguments, and how the curves y=x^⍺ and y=x^β are situated in relation to each other for various pairs of ⍺ and β (with some cool interactions between power functions and exponential functions); you will also get a glimpse into the world of derivatives, to illustrate the problem of monotonicity of y=x^⍺ for any real non-zero ⍺ and positive arguments *x*, and to explain the cusps and rounding in some graphs.

S5. Exponential functions, their properties and graphs

You will learn: about exponential functions f(x)=a^x with a>1, and f(x)=a^x with 0<a<1, and their various properties; by now you know how to plot these functions, and now you will perform transformations of the well-known graphs for plotting new functions: g(x)=f(x)+c, g(x)=f(x+c), g(x)=cf(x), etc.; you get a long problem-solving session in which various properties of exponential functions will be used; we will also illustrate some interesting interactions between exponential functions and power functions.

S6. Important properties of strictly monotone functions

You will learn: about important properties of strictly monotone functions, which will help us understand exponential functions, logarithmic functions as inverses to exponential functions, and solve exponential and logarithmic equations.

S7. Logarithmic functions as inverses to exponential functions

You will learn: the definition and properties of logarithms with various bases; properties of logarithmic functions, their graphs, and graphs of some related functions obtained by transformations of graphs of basic logarithmic functions.

S8. Exponential equations and inequalities

You will learn: how to solve exponential equations and inequalities, starting with some simple ones, ending with some more complex examples; determining inverse functions by solving exponential equations.

S9. Logarithmic equations and inequalities

You will learn: how to solve logarithmic equations and inequalities, starting with some simple ones, ending with some more complex examples; determining inverse functions by solving logarithmic equations.

S10. Applications of exponential and logarithmic functions

You will learn: changes in percent, and the change factor in growth and decay; compound interests and how annuities relate to geometric series; some applications of exponential and logarithmic functions, for example for analysis of growth or decay; logarithmic scale. This section is different than the earlier sections, because you will mostly read about the topics (which are very language intensive) from the Precalculus book; I will introduce each topic in the videos, and you are welcome to ask questions on QA (under the corresponding videos, so that all the other students can find them on the right place) if you need my assistance.

S11. Some more advanced topics

You will learn: some more advanced topics concerning the subjects of the course, like some examples from Calculus (hyperbolic functions; how to demonstrate with help of derivatives that the graphs of exponential functions look like they do; Taylor polynomials for f(x)=e^x; some examples with ODE modelled for situations of growth or decay); you will also get to see some more advanced problems which didn't match any of the categories in the previous sections (like mixed equations and inequalities, i.e., mixtures of radical, exponential, and logarithmic equations).

S12. Extras

You will learn: about all the courses we offer. You will also get a glimpse into our plans for future courses, with approximate (very hypothetical!) release dates.

*Make sure that you check with your professor what parts of the course you will need for your final exam. Such things vary from country to country, from university to university, and they can even vary from year to year at the same university.*

*A detailed description of the content of the course, with all the 222 videos and their titles, and with the texts of all the 239 problems solved during this course, is presented in the resource file *

*“001 List_of_all_Videos_and_Problems_Precalculus_4.pdf” *

*under video 1 ("Introduction to the course"). This content is also presented in video 1.*

## Who this course is for:

- Students who plan to study Algebra, Calculus or Real Analysis
- High school students curious about university mathematics; the course is intended for purchase by adults for these students
- Everybody who wants to brush up their high school maths and gain a deeper understanding of the subject
- College and university students studying advanced courses, who want to understand all the details (concerning exponentials and logarithms) they might have missed in their earlier education
- Students wanting to learn exponentials and logarithms, for example for their College Algebra class.

## Instructors

I am a multilingual mathematician with a passion for mathematics education. I always try to find the simplest possible explanations for mathematical concepts and theories, with illustrations whenever possible, and with geometrical motivations.

I worked as a senior lecturer in mathematics at Uppsala University (from August 2017 to August 2019) and at Mälardalen University (from August 2019 to May 2021) in Sweden, but I terminated my permanent employment to be able to create courses for Udemy full-time.

I am originally from Poland where I studied theoretical mathematics and got pedagogical qualifications at the Copernicus University in Torun (1992-1997). Before that, I enjoyed a very rigorous mathematical education in a mathematical class in high school "Liceum IV" in Torun, which gave me a very solid foundation for everything else I have learned and taught later.

My PhD thesis (2009) was at Uppsala University in Sweden, with the title: "Digital Lines, Sturmian Words, and Continued Fractions".

In 2018 I received four pedagogical prizes from students at the Faculty of Science and Technology of Uppsala University: on May 13th from the students at the Master Program in Engineering Physics; on May 25th from the students at the Master Program in Electrical Engineering; on December 20th from the students at the Master Program in Chemical Engineering; on January 10th 2019 from UTN (Uppsala Union of Engineering and Science Students at Uppsala University).

I speak Polish, Swedish, English, Dutch, and some Russian; learning Ukrainian.

I have a background in medicine and software development. I've done enough mathematics to at least follow along in Hania's courses and I'm learning a lot as I edit the material. I have also written a book about medical software design as it pertains to the medical record ("Rethinking the electronic healthcare record"). For Hania's math courses, it's my job to set up the environment and produce the final output that goes into these courses.