Notes
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Calculator Starter Exercises here and Arithmetic
Review Problems in Volume 2, Three Skills for Algebra, offer numerical experience and setting for the following lessons. They
also provide a chance for teachers and tutors to check arithmetic skills.
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Square Root Simplication - a prequel. Calculators may
compute square root of a number exactly or
approximately. But for the mastery of algebraic reasoning, approximations are
to be avoided. Conventions for simplifying square roots of whole numbers
based on their factorization or prime number decomposition are given
here. Use of these conventions is cosmetic. But use leads to a common
or standard form for expressions involving square roots. The generalization
to cube and further roots of whole numbers is obvious.
Examples
here show how to simplify or represent square and cube roots of whole
numbers with and without the aid of prime number factorization
of the latter. The simplification here may be a cosmetic convention
or fashion in mathematics that leads students and teachers to answers
of the same form, the so-called simplified form.
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Natural Logarithms and Exponentials _ Basic Properties.
The basic or fundamental properties of natural
logarithms and its inverse, the exponential function, provide a framework
for the calculation of of further logarithms and exponentials in
this lesson, and for the calculation of roots and powers with
fractional and real exponents in following lessons. All the foregoing
provide a base for and several interchangeable, that is equivalent, growth
and decay models. This lesson assumes values of natural logarithms and its inverse exponential
function may be given by electronic calculators. The lesson ends with an
a geometric definition of the natural logarithm, a definition or account taken
from Volume 3, Why Slopes and More Mathematics, chapter 19.
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Natural Logarithm Calculator Exercises. They offer more numerical experience
with natural logarithms. Do them.
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Formulas for Even and Odd Roots with Logarithms. After a short review of
roots without logarithms, this lesson gives formulas for the
calculation of even roots of positive numbers and odd roots of real numbers.
Formulas employ the natural logarithm, its inverse - the exponential
function, and in the case of odd roots, the sign function.
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Formulas for Even and Odd Roots with Logarithms - Derivation. discusses
and derives the formulas given in the previous lesson. The domain of
definition of the corresponding formulas defines the domain of the corresponding
function.
Formulas for Fractional Exponents with Logarithms.
This lesson derives formulas for raising number to rational powers.
Formulas again employ the natural logarithm, its inverse - the exponential
function, and in the case of odd roots, the sign function. Again, the domain of
definition of the corresponding formulas defines the domain of the corresponding
function.
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Formulas for Real Exponents with Logarithms. Continuous
extension of formulas for positive number c raised to fractional exponents
implies a formula for real exponents. Formulas for more exponentials
functions result. The latter are inverse functions to logarithms
to base c with c = 10 providing the common logarithm. This lesson describes all and how.
Why the exponential function exp(x) equals ex is explained here.
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Exponential Growth and Decay Models. This lesson on exponential growth
and decay models provides an unified algebraic
and view of discrete and continuous growth
and decay models - discrete compound, continous compound, half-life
and doubling time models. All are or can be expressed in terms
of the natural logarithm and its inverse, the exponential. But half-life
and doubling-time models may be expressed or analysed with the aid
logarithms to base 2 and the 2x exponential function. Which logarithms
and exponentials are employed in the forward and backward use of these
models is a matter of taste, guided by ideas of what is simplest.
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Growth and Decay Models in Biology offers numerical exercises
in the forward and backward use of the compound growth and decay
model A = P[1+ii]m. The backward use employs m-th roots
and logarithms. The numbers in the exercises are illustrative only.
In radioactive decay models, exponential decay continues indefinitely. But
in other exponential growth and decay models, the time interval or period
of validity is not certain. So the models should not be taken as absolute.
The Canadian government exponential model of air traffic growth, beyond
criticism because it was a mathematical formula, led to the expensive
1975 opening of an oversized airport, Mirabel North of Montreal. Around
the same, computer print-outs. Models do not lie, but the assumptions
and data that go into to them may be faulty. In applying mathematics,
calculations doable in a repeatable and reproducible manner, in what becomes
routine, are more trustworthy than models that are used in a one-off, once
only situation. Beware of hope in mathematical formulas that goes
beyond reason or routine practice. It is ok to hope, but verify first
before any firm decision. Good luck.
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