Ch11+Exp+&+Log+Functions

Chapter Eleven - Exponential and Logarithmic Functions

 * ===[[file:Ch11 Syllabus.pdf]]===

>
 * [[file:Exponent Properties.pdf]] summarizes several exponent relationships about which you might want to refresh your memory.
 * [[file:Ch11 Highlights.pdf]] summarizes key points about exponential and logarithmic functions.

> For any real number b >= 0 and any integer n > 1, > //b//^(1///n//) = //n//-root (//b//). > This relationship also holds when //b// < 0 //and// //n// is odd. > **// 11-2A Graphing Calculators: Graphing Exponential Functions //**
 * 11-1 Rational Exponents**
 * Definition of //b//^(1///n//):

11-2 Exponential Functions

 * === [[file:Annuity Examples.pdf]] ===

11-3 The Number //e//

 * //The number **e** is a special value. It happens to be an irrational number, so its value cannot be expressed exactly in decimal form, but it is approximately 2.718.//
 * //**e** is the base of the so-called natural logarithms. Like the number pi (3.1415...), the number **e** appears in many relationships in mathematics, physics, chemistry, engineering, statistics, and more.//
 * //Scientific calculators have built-in functions that use **e.**// The function //**e**// //**x**// //is one; the other is **ln** (which means log// e //). Be sure you are familiar with the use of these special keys.//

Quiz 11-1 to 11-3

 * Here is a set of problems similar to those on the upcoming quiz:
 * [[file:Review 11-1 to 11-3.pdf]]


 * And here is the much anticipated solution to those 11-1 to 11-3 review problems. No peeking until you have tried them yourself!
 * [[file:Review 11-1 to 11-3 solution.pdf]]

11-5 Common Logarithms

 * Base-10 logarithms are often referred to as "common" logarithms, probably because these are the best known and most used. Several scientific scales or units are derived from base-10 logarithms, including chemistry's pH, seismology's Richter scale, and the decibel scale in acoustics.
 * Your calculator has a key devoted to base-10 logarithms, labeled "log". (The "second" function for that same key is likely to be the inverse function 10 x .) If ever you encounter a mathematical expression that includes "log" without a specified base, you may safely assume that the logarithm is base-10.
 * You learned the multiplication tables in elementary school, and perhaps you learned squares of a few integers in middle school. Now it's time to learn the common logarithms of the first few integers:
 * log 1 = 0
 * **log 2 = 0.30103**
 * log 3 = 0.477
 * **log 4 = 0.60206**
 * log 5 = 0.699
 * log 6 = 0.778
 * log 7 = 0.845
 * **log 8 = 0.90309**
 * log 9 = 0.954
 * log 10 = 1


 * The logs for 2, 4, and 8 are given to five decimal places because they're easy to remember and you get greater accuracy. Notice anything special about this group of three?
 * Also, what do you get if you add log 2 to log 3? Do you see that sum elsewhere in the list? Why?
 * What do you get by doubling log 3? Do you see that double in the list? Why?
 * You do not actually have to memorize all these logs. Focus on 2, 3, 5, and 7 (prime numbers). You can get others by combinations of these four.
 * Wait! You need not memorize log 5, because 5 = 10/2, so log 5 = log 10 – log 2 = 1.000 - 0.301 = 0.699 (approximately).
 * Armed with these few common logarithms you can perform a great variety of mental computations or estimations of fairly complex mathematical expressions.

11-6 Exponential and Logarithmic Equations

 * An equation that involves an unknown as an exponent can be addressed using logarithms.
 * Ex: 4^(2x+1) = 64
 * Take the Base 4 log of both sides to get (2x+1) = log 4 64 = (log 64) / (log 4) = 3, so
 * 2x+1 = 3 --> x = 1.
 * So taking logarithms of both sides turns an exponential equation into a more tractable algebraic equation.


 * An equation that involves logarithms might be addressed by using the equivalence principle: Two logarithms will be equal only if their arguments are equal:
 * Ex: log 5 (6x–3) = log 5 39 --> 6x–3 = 39 --> 6x = 42 --> x = 7.

11-7 Natural Logarithms

 * Natural logarithms are those that use the transcendental number //e// as the base. The value of //e// is approximately 2.71828.
 * The expression //**log e x**// by convention is written as //**ln x**//. Use the //**ln**// function on your calculator to obtain the natural logarithm of a positive number.


 * Quiz - Sections 11.4 through 11.7**

[This file will be posted shortly before the quiz is to be taken. -JRH]
 * Quiz 11-4 to 11-7.pdf** is the take-home quiz for Sections 11.4 through 11.7.
 * You may use calculators on all questions except Nos. 1 and 2.
 * Your work on this take-home quiz should reflect your own knowledge of the material.
 * Please limit yourself to no more than 50 minutes.
 * You must show your work for each problem. Use extra paper if you need more space, but please attach any such work to your quiz.


 * Chapter 11 Test Review**
 * [[file:Ch11 Review.pdf]] has a set of practice problems for the chapter test.
 * [[file:Ch11 Review Solution.pdf]] shows the answers to the practice problems.