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If log a (base b) = 4 , log b (base c) = 3 and a = 32c^2, find a,b,c.
Solution
log a (base b) = 4
b^4 = a ....... Eqn *
log b (base c) = 3
c^3 = b ........eqn **
a = 32c^2
From eqn *
b^4 = a, where a =32c^2
b^4 = 32c^2 ....... Eqn 1
Fron eqn **, b= c^3
Subt. This into eqn 1 above,
(C^3)^4 =32c^2
C^12/c^2 = 32
C^10 = 32
C = (2^1/2) = root 2
Put root for c in a= 32c^2
a = 32 *2 = 64
Put 64 for a in eqn *
b^4 = a
b^4 = 64
b = root 8.
Therefore, a = 64, b= root 8 and c = root 2

1. To prove that anything raised to a power of zero is one
2. To prove that 0! = 1

1.
a^x * a^y = a^(x+y)

Let x= 0.

a^0 * a^y = a^o+y

a^0 * a^y = a^y

a^o = a^y /a^y.

a^0 = 1.

2.
Recall that n! = n(n-1)(n-2) * ....*3*2*1
n! = n(n-1)!
If we put n=1.
Then, 1! = 1(1-1)! = 1.0!
So in our factorial notation, de define 0! =1.

Euclid of Alenxandria
Euclid (yōˈklĭd) [key], fl. 300 B.C., Greek mathematician. Little is known of his life other than the fact that he taught at Alexandria, being associated with the school that grew up there in the late 4th cent. B.C. He is famous for his Elements, a presentation in thirteen books of the geometry and other mathematics known in his day. The first six books cover elementary plane geometry and have served since as the basis for most beginning courses on this subject. The other books of the Elements treat the theory of numbers and certain problems in arithmetic (on a geometric basis) and solid geometry, including the five regular polyhedra, or Platonic solids. A few modern historians have questioned Euclid's authorship of the Elements, but he is definitely known to have written other works, most notably the Optics. The great contribution of Euclid was his use of a deductive system for the presentation of mathematics. Primary terms, such as point and line, are defined; unproved assumptions, or postulates, regarding these terms are stated; and a series of statements are then deduced logically from the definitions and postulates. Although Euclid's system no longer satisfies modern requirements of logical rigor, its importance in influencing the direction and method of the development of mathematics is undisputed. One consequence of the critical examination of Euclid's system was the discovery in the early 19th cent. that his fifth postulate, equivalent to the statement that one and only one line parallel to a given line can be drawn through a point external to the line, can not be proved from the other postulates; on the contrary, by substituting a different postulate for this parallel postulate two different self- consistent forms of non-Euclidean geometry were deduced, one by Nikolai I. Lobachevsky (1826) and independently by János Bolyai (1832) and another by Bernhard Riemann (1854). See D. Berlinski, The King of Infinite Space: Euclid and His Elements (2013). The Columbia Electronic Encyclopedia, 6th ed. Copyright © 2012, Columbia University Press. All rights reserved. More on Greek mathematician Euclid from Infoplease: geometry: Types of Geometry - Types of Geometry Euclidean geometry, elementary geometry of two and three dimensions (plane and ... number theory - number theory number theory, branch of mathematics concerned with the properties of the integral

THALES OF MILETUS
Birth
Thales, a Greek mathematician cum
philosopher lived in Pre-Socratic times
around 620-625 BC. He is commonly
known as Thales of Miletus as he
belonged to Miletus in Asian region. He
is also acclaimed to be one of seven
Greek sages that left invincible impact
upon philosophy and other sciences.
Thales unveiled an era not only in world
of science but also in philosophy
that’s why he has been dubbed as
“father of science”.
Contribution
He made important contribution to
astronomy, mathematics and
philosophy as Bertrand Russell said
“western philosophy begins with
Thales”. Aristotle also described him
as the first philosopher among Greeks.
He not only influenced the later
philosophers but also played important
role in development of philosophy. He
initiated a new tradition that is
negation of previous influence of
mythology in providing an explanation
of the natural phenomena or existence.
This altogether brought about
revolution in philosophy. Instead of
conventional ideas, he advanced
scientific method by setting forth
hypothesis and then identified the
underlying principles. He influenced
later Greek thinkers and philosophers
such as Pythagoras and others. His
essential belief was as Laertius said
“Water constituted the principles of
all things”.
As Cicero said this about Thales:
“Thales assures that water is the
principle of all things; and that God is
that mind which shaped and created all
things from water”
He also brought up new rational basis
of things rather than initially held
supernatural explanations. One such
example was his attempt to explain
earthquakes by movement of earth or
floating of earth on water rather than
attributing it to some supernatural
incidence.
Contribution in Mathematics
In regards to mathematics, he reached
at solution of problems through
geometry for instance the height or
distance was calculated by him through
geometry. He also used the same
scientific method that is method of
deduction and reasoning rather than
alluding to mythology. Thus he was a
pioneer in discovering new scientific
method in mathematics that’s why he
is regarded as the first true
mathematician. Another remarkable
achievement of his was deriving the
theorem, popularly known as Thales
theorem on basis of deductive
reasoning.
His perception of geometry can be
realized from his strongly held notion
that “space is the greatest thing as it
contains all things”
With regards to geometry, his approach
was much more scientific and
theoretical. Most of times, he made use
of his knowledge in practical ways for
instance he utilized his knowledge of
right and similar triangles in finding the
height or length of pyramids. Besides
this, he also invented his own theorem,
which is related with inscription of
triangle in circle and circle’s diameter
is equivalent to one leg. In addition to
this, he is also attributed to establish
another theorem often called as
intercept theorem. He is known to
make discovery regarding circle that if
cut half by its diameter and that base
angles as well as vertical angles of
triangle are equal.
Thales theorem:
Thus he brought new vistas to existing
ideals of mathematics and philosophy
mainly by his attempt to rediscover it in
more practical or theoretical terms.
Death
Although the exact date of his death is
not known but it is presumed that he
died between 547-546 BC

Its another day in a new month!
Even if 2day would come 2 meet u alive then ur success will surely come true: even if 2day would also go and vanish, then every trace of violence or iniquity shall also evaluate ur life. Has every leaves open their surface to nature and receive their glorys of living from their CREATOR during the day, so broadly shall ur heart open too to receive ur glory 4rm d CREATOR. Anything of hindrance no matter how inevitable, there shall be a departure betw u by above mercys' and grace. Learn hw to live ur life in d pure form and everything u wish would be solved as if its already been written. 2day is history of morrow and ensure it turn out to be a great ones' when the time comes. Discover urself on everyday just like me "dy'howner" who uses all his day and night to the study of mathematics and explode to loving all under roof and let make this time of ur life worthliving as peace,cool, admire,innovative , love, happy, discovery ............., may dis not be ur last day on earth.
dy'howner pray