YOU are better than YOU think. Show yourself how:
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Learn to read notes and textbooks like a lawyer, so that no nuance, no subtlety and no clause escapes your attention. Read logic
chapters 1 to 5 in online volume Three
Skills for Algebra for greater skills & confidence in
work |
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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;
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Caution: Site advice is approximately correct, for some circumstances, not all. That leaves room for thought |
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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
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For online automated help in senior high school maths & calculus,
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With overlap, each site quickmath
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services, some free, some not, all based on webmathematica. Good luck.
More Algebra Hints
7 Natural Logarithms and Exponentials
The natural logarithm ln(x) is defined for x > 0. The exponential function exp(x) is defined for all real x.
Uniqueness (or 1 to 1) Property: If a > 0, b> 0 and ln(a) = ln(b) then a = b.
Inversion Properties
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ln(exp(x)) = x for all real x
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exp(ln(x)) = x if x > 0
For each real number a, x = exp(a) is the unique solution of a = ln(x). Solving the latter equation is one way to define or compute exp(a).
Fundamental property of logarithms ln(ab) = ln(b) +ln(a) (proof available in calculus)
Fundamental property of exponentials: exp(x1) · exp(x2) = exp(x1+x2) This follows from the uniqueness property of logarithms and the fundamental properties of logarithms.
The fundamental property of logarithms implies
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ln( 1/a) = (-1) ln(a) as 0 = ln(1) = ln ( (1/a) a )
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ln(a m) = m ln (a) for all whole numbers and then for all integers. integers.
Logarithms to base c > 0.The logarithm of x > 0 to a base c > 0 is given by
The logarithm of x > 0 to a base 10 is given by
The button log(x) on a calculator computes log10(x). The definition of logc(x) in terms of ln(x) implies
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Roots and rational powers of positive numbers
How to compute using logs and exponentials
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ln(a m) = m ln (a) implies a m = exp( ln(a m)) = exp(m ln (a)). Eg (1.7)3 = exp( 5 ln(1.7)).
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Now b = a 1/m when and only when b m = a. The latter implies ln(a) = ln(b m) = m ln (b) and hence ln(b) = (1/m) ln (a). So b = exp (ln(b)) = exp( (1/m) ln(a) ) = a 1/m Eg. 8 1/3 = exp( (1/3) ln(8)
Now if m and n have no comon divisors, and n is nonzero, let the m/n power of a m/n = (a m)1/n
Then a m/n = (a m)1/n = exp( (1/n) ln(a m)) = exp( (1/n)m ln(a m)) = exp( (m/n) ln(a))
Roots and rational powers of positive numbers
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EG: 8 1/3 = exp( (1/3) ln(8)
EG 8 2/3 = exp( (2/3) ln(8)) = ( 8 1/3)
2
Exponentials of Real Numbers a x = exp( x ln(a))
For x = m/n and a > 0, a x = a m/n = exp( (m/n) ln(a)) = exp( x ln(a)). This suggests putting a x = exp( x ln(a)) for x irrational. Then
a x = exp( x ln(a)) for all real x for a > 0
and not only for rational numbers. From this definition, ln a x = x ln(a). Therefore loga(a x) = x because loga(x) = ln(x)/ln(a).
Properties of Exponentials
Now (a x)y = exp(y ln(a x )) = exp(y x ln(a )) = a yx = a xy Therefore
(a x)y = a xy (Exponential of an exponential)
Now a xay = exp(x ln(a)) · exp(y ln(a) = exp(x ln(a)+y ln(a)) = exp( (x +y )ln(a) ) = a x+y Therefore has the exponential property
a xay = a x+y for all real numbers x and y when a > 0.
Now for the natural number e = exp(1) = 2.718281828... (irrational, deci), the natural logarithm of e, ln (e) = 1 Therefore
e x = exp( x) for all real x when a > 0
as a x = exp( x ln(a)). Calculators often have a button marked e x for the evaluation of the exponential function exp( x)
Caution: the capital EXP on some calculators will not help you with the calculation of exp(x). Use the button marked e x instead.
Even Roots of Roots Numbers
Here x 2 > 0 for all real numbers x. Therefore the equation x 2 = b only has solutions x when b > 0, that is only when b is non-negative. Defining
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as the nonnegative real solution of x 2 = b works only if b is positive. This solution is given by a ½ = exp( ½ln(b)). See above.
Similarly, if n = 2m > 0 is an even, then x n = x 2m > 0 for all real numbers x. So the equation x 2m = b only has solutions x when b > 0, that is only when b is non-negative. The foregoing implies defining
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nÖbas the nonnegative real solution of x 2m = b works only if b is positive. This solution is then given by a1/n = exp( (1/n)ln(b)). See above.
Odd Roots of Real Numbers
Each real number x = sign(x) |x|. For instance
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+5 = (+1) 5 as sign (5) = +1 and |+5| = 5 = distance of +5 = 5 to origin 0
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-4 = (-1) 4 as sign (4) = -1 and |-4| = 4 = distance of -4 to origin 0
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0 = (0)(0) as sign(0) = 0 and |0| = 0 = distance of 0 to itself.
Now sign(x) = +1, 0 or -1. In all, three cases [sign(x)]2 = 1. Therefore
x3 = [sign(x)]|x|3
The equation x3 = b = sign(b) |b| has one and only real solution real solution, namely x = sign(b) exp( (1/3) ln(|b|) ) as the horizontal line y = b intersects the graph of y = x3 at most one point. Exercise: Sketch the graph of y = x3 for -2 < x < 2. For each nonzero real number b let b1/3 and
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= sign(b) exp( (1/3) ln(|b|) ) |
is the real solution of x 3 = b. Let
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= 0 |
Similarly, if n = 2m+1 > 0 is an even, then x 2m+1 = sign(x) |x|2m+1 For each nonzero real number b let b1/n and
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nÖb= sign(b) exp( (1/n) ln(|b|) ) be the real solution of xn = b . Let
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nÖ0= 0
Real roots and exponentials for complex numbersEach nonzero complex number z = |z| cis(q) for some angle q with say 0 < q < 2p = 360 degrees. Put
Then z a+ b = z a z b. Now x = z 1/n = cis(q/n) exp((1/n) ln|z|) is the so-called principal complex valued solution of the equation x n = z. (z given). But if z = b is real. Then z = |b| cis(0) or z = |b| cis( 180 degrees) with |z| = b in both cases If b > 0, then z 1/n = cis(q/n) exp((1/n) ln|z|) ) = exp((1/n) ln(b)). But if b < 0, then z 1/n = cis(180/n degrees) exp((1/n) ln|b|) ) lies on the ray with angle 180/n degrees in the first quadrant of the complex plane. This complex root does not belong to the real number line. For n odd and b < 0, it differs from the real solution x = sign(b) exp( (1/n) ln(|b|) ) of xn = b. Now x = z 1/n = cis(q/n) exp((1/n) ln|z|) is a complex valued solution of the equation x n = z = b. But the latter equation has n solutions given by the formula x = cis( (q +360 k degrees)/ n) exp((1/n) ln|z|) where 0 < k < n. When z = b < 0, the real solution or root x = sign(b) exp( (1/n) ln(|b|) ) of xn = b is not the principal complex valued solution. In the complex plane, the presence of n solutions of x n = z leads to some choice in defining or selecting the principal value of z 1/n |
Definition of Natural Logarithms
What the ln(x) button computes for x > 0.
The next two diagram show the area-based definition of the natural logarithm ln(a) or ln(b) in the two mutually exclusive cases a > 1 and 0 < b < 1. Note: The graph of y=ln(x) is different from the graph of t = 1/s. The latter is used to define ln(x), not graph it.
For a ³ 1, the value of ln(a) is given by the area from s = 1 to s = a under the curve y = [1/(s)]. Here we take or assume ln(1) = 0. It can be shown that ln(a) ® 0 when when a approaches 1 through values above or greater than 1. Observe increasing a increases the area under the curve = ln(a).
For 0 < b < 1, the value of ln(b) is given by (-1) times
the area under the curve y = [1/(s)] from s = b to s
= 1.
Note: The graph of y=ln(x) is different from the graph of t = 1/s. The latter is used to define ln(x). Exercise: Find the graph of ln(x) in a text book.
www.whyslopes.com
Algebra, Odds & Ends,
1. Hints for Exams 2A. Exact Arithmetic 2B. Fractions Briefly 3. What is a Variable? 4.. Square Roots 5. Straight Lines 6. Problem Solving Methods 7. Trig and Complex No. 8. Complex Applet 9. History of No.s 10. ln(x) and exp(x) 13. Rename the > Sign 14. Problems: Quadratics 15. Problems: Algebra Test 16. Problems: Linear Eqns I 17. Problems: Linear Eqns II 18. Problem Solving Hints 19. Functions & Sets 20. Independent Variables 21. Why Logic 22. Why Math 23. The 15 Times Table 24. The 20 Times Table 25. Algebra Formulas 26. On Learning Maths 27. Maths in Biology 28. Navigation +Time 29 Quibble-What is Algebra 30. Logic in Maths
Odd and Ends, Essays
Constant Retirement Rate Road Safety 3 Strikes Law in California. Math HOW-TOs 9 Steps in Maths
Study With Others: twiddla.com has developed a collaborative whiteboard with audio & text chat that allows students, tutors & teachers to explore & scribble on blank pages and copies of webpages together, If scribbling is awkward with one browser, try another.
In Volume 2, Three Skills for Algebra, Chapters 8 to 14 and postscript What is a Variable point to a greater & clear use of words in algebra. Chapter 14 introduces a 4th skill for algebra, an elaboration of the third: - The direct and indirect use of formulas, numerically and algebraically, is unifying theme that should be mentioned aloud, with words, in each and every use of formula.