About Exponents
and Logarithms
TABLE OF CONTENTS
About Exponents and Logarithms .................................................................................. 1
What is an EXPONENT?................................................................................................ 1
What is a LOGARITHM?............................................................................................... 1
Power Function ................................................................................................................. 1
Squares ............................................................................................................................ 1
Calculator Usage ......................................................................................................... 2
Cubes............................................................................................................................... 2
Square/Cube Roots ........................................................................................................... 2
Square Roots ................................................................................................................... 2
Cube Roots...................................................................................................................... 3
Exponents .......................................................................................................................... 4
Exponents........................................................................................................................ 4
Properties ........................................................................................................................ 5
Exponential Functions...................................................................................................... 6
Exponential Functions .................................................................................................... 6
Logarithms ........................................................................................................................ 7
Logarithms ...................................................................................................................... 7
Logarithmic Properties.................................................................................................... 8
Changing Bases............................................................................................................... 8
Natural Logarithms .......................................................................................................... 9
Natural Logarithms ......................................................................................................... 9
Glossary ........................................................................................................................... 10
References........................................................................................................................ 13
About Exponents and Logarithms
What is an EXPONENT?
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An exponent is an indicator of the number of times a value multiplies
itself, denoted by a superscript number placed on the right of the
value.
●
Exponentiation is used in a wide-range of fields, from mathematics to
biology.
What is a LOGARITHM?
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A logarithm is the inverse of an exponential function.
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An understanding of logarithms is fundamental to a large number of
scientific concepts.
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In mathematics, a logarithm is the power to which a base must be
raised to give a specified value (e.g. log28, is 3, as 23 = 8).
Power Function
Squares
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The square of x is written as x². It is called a square because it
represents the area of a square with a side length of x. As such, when
you multiply a number by itself, you get the square of the number.
This value will always be non-negative.
●
You should know the squares of the integers from 1 to 10. You should
also be able to square numbers consisting of an integer followed by
zeroes. To do so, simply square the leading integer, then double the
number of zeroes on the end.
●
It is possible to square fractions, as well, by simply squaring both the
numerator and the denominator.
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You can also square numbers with decimal points. Similar to numbers
with trailing zeroes, for decimals, there are double the number of
terms after the decimal point in comparison with the initial value. The
squaring of the term itself is equivalent to squaring the term without
regarding the decimals, then accounting for the decimals afterwards.
Calculator Usage:
●
To square a number with a calculator, look for a button marked x2.
Sometimes this is printed above the key rather than on the key. If this
is the case, you will have to press some other button first. It might be
labeled INV or 2ND. Most calculators will colour-code the buttons, so
you can find the right one easily.
IMPORTANT: When you square a number with many digits
following the decimal point, using a calculator, you will get an
approximation rather than an exact answer. For example, a calculator
which displays 10 digits will tell you that 1.000000082 = 1.00000016
when it actually equals 1.0000001600000064. Also, when you square
numbers, they often become very large or very small, so your
calculator will display them using scientific notation.
Cubes
●
The cube of x is written as x3. It is called a cube, because it is the
volume of a cube with a side length of x. The cube of a number is
equivalent to the number times its square, which can be either positive
or negative.
Square/Cube Roots
Square Roots
●
The square root is the inverse function of the square function, written
as .
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Being the inverse means that if a2 = b, then pb = a. Geometrically, the
inverse of a function is its reflection across the line y = x. The domain
of a function is the range of its inverse, and the range of a function is
the domain of its inverse.
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So, since the square of a number is always positive (A positive
number times a positive number is positive, and a negative number
times a negative number is positive), the range of the square function
is all positive real numbers and zero; hence, the domain of the square
root function is all positive real numbers and zero.
●
Recall, you cannot take the square root of a negative number solely
using real numbers.
●
Every number has two possible square roots, since:
a² = (-a)².
Cube Roots
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The cube root function is the inverse of the cube function, written
as
.
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Cubes can be negative, so you can take the cube root of a negative
number, yielding one real root.
Exponents
Exponents
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The concept of squares and cubes can be generalized to the
multiplication of a term by itself any number of times.
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We say "a to the x" or "a to the power of x," when referring to an
exponential term. If the exponent is negative, we say "a to the
negative x" or "a to the power of negative x."
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The addition of exponents corresponds to multiplication, as shown
below, while the subtraction of exponents corresponds to division, in
which the exponent of the denominator is subtracted from the
exponent of the numerator. Note that all terms used must have a
common base.
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As illustrated below, any term to the power of zero is equal to 1.
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The following illustrates properties of rational exponents.
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Where r is rational, and x is irrational, the following holds.
Properties
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Exponential Functions
Exponential Functions
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An exponential function is a function of the form y = ax, where a is
any positive real number.
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An example of an exponential function is the function y = ex, where e
is Euler's number, an irrational number approximately equal to
2.718281828. It is sometimes written as y = exp(x). It is important to
distinguish exponential functions from power functions (like square
and cube), which are of the form y = xn, where n is a real number.
Exponential functions have a constant base and a variable exponent;
power functions have a variable base and a constant exponent.
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Exponential functions only take on positive values.
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If a > 1, then the function goes to infinity as x gets larger and to 0 as x
gets smaller.
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If 0 < a < 1, then the function goes to 0 as x gets larger and to infinity
as x gets smaller.
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All exponential functions pass through the point (0,1).
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A value that grows according to an exponential function is said to
exhibit exponential growth. In colloquial speech, this often just means
that something is growing fast. Don't confuse this with the precise
mathematical definition. If a is just a little bigger than 1, the growth
isn't actually that fast.
A good way to think of it is: something has exponential growth if it
grows faster the longer it has been growing.
Logarithms
Logarithms
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Logarithmic functions are the inverses of exponential functions.
This means that a logarithm function is the reflection of an
exponential function across the line y = x. This tells us some
important facts:
► Logarithm functions always pass through (1,0).
► Logarithm functions are only defined for positive real numbers.
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► For base a > 1 the value of a logarithm gets smaller as x gets closer
to 0 and larger as x gets larger.
► For 0 < a < 1 the value gets smaller as x gets larger and larger as x
approaches 0.
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y = logax if and only if ay = x.
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Logarithms are useful for measuring quantities that vary over a large
range of values.
Logarithmic Properties
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Property 1: logaMN = logaM + logaN
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Property 2: logaM/N = logaM - logaN
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Property 3: logaMN = N x logaM
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Property 4: loga1 = 0
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Property 5: logaa = 1
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Property 6: logax = logay, if and only if x = y
Changing Bases
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Scientific calculators with log buttons do not provide a way to enter
the base. Why is this? It is partly due to the fact that there are two
bases which are used more commonly than any others: base 10 and
base e. Another reason is that it is easy to convert from one base to
another. The formula is as follows:
logold basenumber = lognew basenumber
lognew baseold base
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Natural Logarithms
Natural Logarithms
●
When people write log without indicating a base, they mean to use
base 10. This is called the common logarithm. Another base
frequently used with logarithms is base e.
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A logarithm using base e is written ln x. This is sometimes
pronounced "lawn x." It is called the natural logarithm.
●
e is Euler's number. It is an irrational number approximately equal to
2.718281828459045235360287471352, and named after Leonhard
Euler (pronounced "oiler"), an eighteenth century Swiss
mathematician.
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One way to define e is with the infinite sum:
e = 1/0! + 1/1! + 1/2! + 1/3! + ....
In this sum, the symbol "!" means factorial:
n! = n x (n - 1) x n - 2 ... 1.
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Another way to define e (using limits) is:
e = limn→∞ (1 + 1/n)n
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Glossary
Cartesian Plane:
is a plane with a rectangular coordinate system that
associates each point in the plane with a unique
pair of numbers in an ordered pair of the form
(x,y).
Co-ordinate:
is a point on the Cartesian plane that has both a
vertical (y) and a horizontal (x) value.
Cube:
what you get when you multiply a number by its
square.
Cube Root:
is the inverse of the cube function.
Decimal:
consists of a whole number to the left of the
decimal point, and a fractional number to the right
of the decimal point.
Denominator:
is the bottom portion of a fraction.
Domain:
all of the valid values of x for a given expression,
written as [ lowest x, highest x ].
Exponent:
is a number denoted by a small numeral placed
above and to the right of a numerical quantity,
which indicates the number of times that quantity
is multiplied by itself.
Exponential Function: is a function of the form y = ax, where a is
any positive real number.
Fraction:
consisting of a numerator and a denominator, is
representative of a relational ratio.
Function:
is a relation, such that each element of a set is
uniquely associated with an element of another set.
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Infinity:
the maximum or minimum (if negative) limit for
all real numbers.
Integer:
meaning the set of whole numbers, coupled with
the negative values of that set.
Irrational Number:
is any number that cannot be written in the form
x/y.
Logarithm:
is the inverse of the exponent function.
Natural Logarithm:
a logarithm using base e and written as ln x.
Numerator:
the top portion of a fraction.
Origin:
the midpoint of the co-ordinate plane, usually
being (0, 0).
Range:
all of the valid values of y for a given expression,
written as [ lowest y, highest y ].
Rational Numbers:
any number of the form x/y, where x can be any
integer, and y can be a non-zero integer.
Scientific Notation:
a system in which a number is expressed as a
number between 1 and 10, times 10 raised to an
exponent.
Square:
what you get when you multiply a number by
itself.
Square Root:
is the inverse of the square function.
Volume:
the amount of space occupied by an object.
x-axis:
known as the horizontal number line that runs left
to right on the Cartesian Plane.
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y-axis:
known as the vertical number line that runs up and
down the Cartesian Plane.
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References
http://en.wikipedia.org
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