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exponents, roots and logarithms | R
2018-10-04 18:11
http://qindex.info/d.php?c=5216#5357

Summary | R
From the beginning the base of the exponent could be any known number. However the exponent was a natural number greater than 1. Thereafter 1, fractions, 0, and negative numbers were allowed to be exponents in order.
http://qindex.info/d.php?c=5216#5217

1 Repeated Multiplication | R
Multiplication is repeated addition. a × 3 = a + a + a Exponents are originally repeated multiplication. a ^ 3 = a × a × a
http://qindex.info/d.php?c=5216#5213

2 Inverse Operator, Root | R
Addition and subtraction are inverse operations. a + b = c; c - b = a; Multiplication and division are inverse operations. a × b = c; c ÷ b = a; Exponents and roots are inverse operations. a ^ b = c; The b-th root of c is a.
http://qindex.info/d.php?c=5216#5214

3 Irrational Number | R
According to Pythagoras' theorem the diagonal length of a square with each side measuring one unit is the square root of 2. The assumption that square root of 2 could be expressed as a ratio of two integers deduces a contradiction.
http://qindex.info/d.php?c=5216#5147

4 The power of 1 | R
a^3 is a×a×a and a^2 is a×a, so define a^1 as a.
http://qindex.info/d.php?c=5216#5207

5 The power of a fraction | R
(a^b)^c = a^(b×c) and a^1 = a, so if we define a^(1/b) as the b-th root of a, then (a^b)^(1/b) = a^(b×(1/b)) = a^1 = a
http://qindex.info/d.php?c=5216#5222

6 The power of 0 | R
The limit of a^x as x approaches 0 from the right is 1. So define a^0 as 1.
http://qindex.info/d.php?c=5216#5215

7 The power of a negative number | R
(a^b) × (a^c) = a^(b+c) and a^0 = 1, so if we define a^(-b) as 1/(a^b), then (a^b) × (a^(-b)) = a^(b+(-b)) = a^0 = 1
http://qindex.info/d.php?c=5216#5224

8 Inverse Operator, Logarithm | R
a ^ b = c An inverse operator only when b is a natural number greater than 2: The b-th root of c is a. Another inverse operator when b is a real number: The log base a of c is b.
http://qindex.info/d.php?c=5216#5226

Logarithmic and exponential functions | R
Very low sound
http://qindex.info/d.php?c=5216#3472

Reading b^n | R
"b raised to the power of n", "b to the n-th power", "b to the n-th", "b to the n", "the n-th power of b", "b squared", "b cubed"
http://qindex.info/d.php?c=5216#5223

Saying Logarithm | R
"The base of the log raised to the power of what's on the other side of the equal sign will equal the number that the log is operating on."
http://qindex.info/d.php?c=5216#5227

Saying Logarithm | R
If you want to find log base b of x, you're asking "What power do I have to raise b to in oder to get x?" Or you can say, "If I raise b to the power of y, I'm going to get x."
http://qindex.info/d.php?c=5216#5228

Triangle of Power | R
Triangle of Power
http://qindex.info/d.php?c=5216#5232
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