Appetizers and Lessons for Mathematics and Reason 
www.whyslopes.com - mathematics as an art and discipline, step-by-step  Parents: See Help Your Child/ Teen Learn 
Français
:
||Définition d'une variable || Algèbre || Arithmetique || Logique ||La raison basée sur les règles et modelés||

Online Volumes (Book Orders)
1,  Elements of Reason.
1A. Pattern Based Reason 
1B. Math Curriculum Notes
2. Three Skills for Algebra
   Three Skills for Algebra
3. Why Slopes & More Math
 Avid Readers: Try Pattern Based Reason  & chs 
 1 to 12, 14,  16 & 17  in  Three Skills for Algebra.
More Site Areas 
1. Help Your Child/ Teen Learn 
2. Solving Linear Equations  
3. Fractions Ratios Rates Proportions, Units
4. Euclidean Geometry
5. Analytic Geometry/Functions 
6. Number Theory
7. Calculus Introduction
8. Complex Numbers 
More Site Areas 
9. Quebec Maths Education  
10. Secondary IV(?) maths
11. Real  Analysis 
12. LaTeX2HotEqn:
13. Electric Circuits Etc  
14. Algebra, Odds & Ends, Etc
16  LAMP - Course re Design Plans
17. Math Education Essays
Teacher-Tutor Info & How-TOs
1. Arithmetic Reference
2. Algebra Starters 
3. More Algebra 
4. Geometry Starters
5. More Geometry
6. Calculus Modifiers 
7. Multiple Logics in Maths
8. Math Ed. Issues

Back ] Area Entrance & Hub ] Next ]


Tutors - All Subjects

AU:  tutorfinder.com.au
CDN :  findatutor.ca 
CDN: .i-tutor.ca
CDN: Montreal Tutors
NZ:   findatutor.co.nz
UK:   tutorhunt.com
USA:   wiziq.com
USA: ziizoo.com

use or become a tutor  at your own risk 


YOU are better than YOU think. Show yourself  how:  

      |      
//  _   _ \\
/\             /\
  <|  (o)   (o)   |> 
 \     | |      / 

Read  logic chapters 1 to 5  in online volume Three Skills for Algebra  for greater skills & confidence in  work 
and study.

Learn to read notes and textbooks like a lawyer, so that no nuance, no subtlety and no clause escapes your attention.

 -/[]\- 
||
   / \_ 
 ||||||||||||||||||||||||||||

 Logic chapters 1 to 5  re- appear not in sequence, as is or longer,  in  Volume 1A,  Pattern Based Reason, Bon Appetite.

Logic Mastery
 Amazing, Amusing, Amorous,  Delicious, Delightful, Edifying, Strengthening Elixir. 
It eases work & learning difficulties Makes the hard easier. Opens eyes. Leads to greater precision.
in reading and
writing

Logic mastery makes the hard, easier. Logic mastery  leads to better, stronger and richer comprehension.  Logic mastery  improves reading and writing.  Logic mastery ease learning difficulties.  Logic mastery gives a headstart.  In sum, logic mastery  will develops critical thinking, improve reading and writing, and give a firmer base for work and studies at many levels. Good luck.


After logic  (a) continue reading Three Skills for Algebra, chapters 8 to 14  and do so alongside site area on solving liinear Equations ; or (b) see this calculus starter lesson and Volume 3, Why Slopes  & More Math, chapters 2 to 6;

      |      
//  _   _ \\
/\             /\
<|   (o)   (o)  |> 
     | |     |
   \             /   
\    =   /

Caution: Site advice is approximately correct, for some circumstances, not all. That leaves room for thought

 -/[]\- 
||
  _ / \     
 ||||||||||||||||||||||||||||

What may be learnt and when depends on how skills and concepts are developed. Making the hard easier and clearer will allow earlier & richer development of skills and concepts.


Try the Twiddla Whiteboard. In principle, it  allows to people to draw and chat together online on a copy of this webpage or a clean sheet. The chat may be via text or audio.  Visit www.twiddla.com to set up whiteboards to work with the webpage of your choice.

For online automated help in senior high school maths & calculus, visit  quickmath.com  For Automatic Calculus and Algebra Help with derivatives, integrals, graphs, linear equations, matrix algebra, visit calc101.com  With  overlap, each site quickmath & calc101offers a different range of services, some free, some not, all based on webmathematica. Good luck.

Chapter 19
Real Logs, Powers and Exponentials

Questions

What does an electronic calculator compute when the natural logarithm button on it is pressed? The answer follows from a long chain of mathematical concepts and reasoning. The definition below of the natural logarithm in terms of area under a curve y = [1/(s)] provides a preview (or review) of notions employed in integral calculus - the subject which treats, in the first instance, the calculation of areas under curves. This chapter assumes a previous acquaintance with logarithms, powers and exponentials.

Electronic Calculators

Electronic calculators allow the illustration and an electronic, pre-programmed confirmation of the basic relationships between logarithms, exponentials and powers. The following computations can be illustrated with an electronic calculator.

FOOTNOTE: This represents or indicates the easy buttons-on-a-calculator approach to the description/explanation of logarithms, exponentials and powers together with the relationships between the calculations invoked by the buttons.


  1. The logarithm of x > 0 to a base a > 0 is given by
    loga(x) = ln(x)
    ln(a)
    ·
    For example on some electronic calculators, log10(6) is giving by pressing the 6 and then the log button (in some order). This should give the same result as computing ln(6)¸ln(10). (Exercise: check this by pushing buttons.)
  2. Multiplying a number a > 0 by itself n times gives an. But the calculation of exp(n ln(a)) gives the same result. So the original definition of an for a > 0 is consistent with the more general definition given for real numbers x by
    ax = exp(x ln(a)).
    For examples, compute 52 and exp(2ln(5)) on an electronic calculator. Also compute the following:
    x
    = 100.2,     y = exp(0.2ln(10)),     
    x
    5 and exp(5 ln(x)).
  3. For a > 0,
    loga(ax) = x
    and for v > 0, u = loga(v) implies au = v. For examples compute log(103)-3 and 10log(8).
The special cases a = 10 and a = 2 are of interest most likely due to the recent historical preference for decimal (base 10) arithmetic and due to, still more recently, to the advent of computers with their binary (base 2) arithmetic. Also of interest is a third case a = e where e is the so-called natural number. See below.

Definitions of logarithms and exponential functions are given in the next two webpages to explain and derive the computational relationships described above.

www.whyslopes.com
Volume 3,  Why Slopes and More Math
-  

Foreword, One Calculus  preview and Online Chapters: (V) signals video (RealPlayer Format)  to watch 

Area Entrance & Hub
Foreword
Chapter Descriptions
1. Introduction
2. Calculus Starter Lesson
2. Second Preview Begins
2 Skier in Motion (V)
2 The Skier (V)
2. Position Dependent (V)
3 Slope & Extrema (V)
4 Single Factor Analysis (V)
4 Two Factor (V)
4 More Factors (V)
4 With Divisors (V)
5 Maxima & Minima Tests
6 Jumps & Discontinuities
8 Review  (optional)
9 On Calculus Studies
11 Slope of Slope
13  Acceleration
14 Limits & Error Control (V)
14 Limit of a Fn.
14. Limited Error Control
14 Signif. Digits
14 Cauchy Limits
14 Sequence Limits
14 Decimal Arith.
15 What is Slope (V)
15 Slope Calculation (V)
15 Slope, a Limit
15 Tangent Lines
15 Linear Approx.,
15 Limits via Algebra (V)
15 Recap.
PS.Chain Rule for Polys
PS Chain Rule- General  (V) -
PS More Chain Rule (V)
PS - Sign Analysis (V)
16 What is Velocity
17  What is Area
18 Integration
18 Area Calculation
18  Fn DefN, 6 Ways
19 Logs & Powers
19 Natural Log.
19 Exponential Fn.
20 What's Next
21 Add Vectors
22 Complex #'s
23 Complex #'s
23 Trig Identity
23 Proofs of.
24 Complex Logs etc

Units in Calculations:
7 Velocity
7 Varying Velocity Example
7. Velocity Calculation
7 Changing Units
7 Same Velocity  Motions
10 Slopes without Units.
10 Units & Slopes
10  Units in Cost vs. Quantity
10  How Units  Appear
10 Unit  Elimination
10 Partial Elimination
10 Interest & Units
12 More on Units
Content Guide

Enriched material: The Appendices of Volume 3 are located in the Real  Analysis  Area.

Pigeon Hole Principle
Constant Difference Thm
Continuous Functions
Rational Functions
Mean Value Theorem
One Side Range Theorem
Range On One Side Theorem
Integration & Lipschitz
 Continuity


These appendices continue the
decimal viewpoint of limits, error
control and continuity begun
in Chapter 14. The One Sided
 Range Theorem
is a postscript,
not in printed version.



Online Volume 2, Three Skills for Algebra, Chapters 1 to 25 - skip 18., verbalizes and explains key skills and concepts, those needed in calculus, again to make the hard easier. A visual understanding of complex numbers may help - serve as back ground info,  in partial fraction decomposition.

 

 


www.whyslopes.com

[Top of this Page] [Site Exit] Back ] Area Entrance & Hub ] Next ]
[Comments, Reactions, Feedback][ Road Safety Message ]
: Favourite SitesBBC News  and mathematics portion of  English National Curriculum  



All trademarks and copyrights on this page are owned by their respective owners.
Copyright to comments & contributions are owned by the Poster. 
The Rest © 1995 onward by site author,   Alan Selby,
a 1983 McGill. Ph. D. in mathematics
All Rights Reserved.