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.
- The logarithm of x > 0 to a base a > 0 is given by
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.)
- 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
For examples, compute 52 and exp(2ln(5)) on an electronic
calculator. Also compute the following:
x = 100.2, y = exp(0.2ln(10)),
x5 and exp(5 ln(x)).
- For a > 0,
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.
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Why Slopes
and
More Math
understanding & explaining
Reason and Math
Volume 3
Printed in Canada
ISBN 0-9697564-3-7
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Content Guide
Foreword
Chapter Descriptions
1. Introduction
Preview -why slopes in 1983
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
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.
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