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Miscellaneous Math Topics
2 Logarithms/Exponents
2.1 Exponents
Exponents are the superscripts on numbers or other mathematical operations.
Exponentiation is sometimes called raising a number to a power . This process
involves two numbers : the exponent and the base. The result is a power of the
base. For example, in the argument a^{n} = x, a is the base, n is the
exponent,
and x is the n^{th} power of a. This is equivalent to raising a to the n^{th}
power.
For example:
Any number raised to the 1^{st} power is itself
and any number raised to the 0^{th}
power is one. For example:
Complex arguments can also be exponentiated and exponents
may themselves
be complex arguments. For example:
The polynomial quotient
is raised to the (X + 1)^{th} power. Raising a
nonzero number to the 1 power is to take the inverse of the number, or in other
words, to produce the number’ s reciprocal . For example:
Note that in this example I factor the components of 2
into 1 and 2 to make
the steps clearer . This is one of the allowable operations on integers :
• Multiplying exponentiated numbers with the same base is done by adding
the exponents: . For example:
. This also works with negative exponents :
. For
example:. Equivalently ,
• Exponentiating an exponentiated number is done by multiplying the ex
ponents: . For example,
. To check
this, note that .
• Exponents can be factored: .
This can be useful:
• Numbers with the same base and same exponent can be added:
and
Exponential functions are often used to model growth and decay because
they multiply at a constant rate. For example, a population that doubled in
size every 25 years could be modeled with an exponential function.
Exercises: Exponents
1. Simplify (x^{2} + 3x) + (2x^{2} + 2x +
5)
2. (2x^{2})(x^{2})
3. Calculate (3^{2})^{2}
4. Calculate
2.2 Logarithms
Like the exponent , the logarithm involves two numbers, the base number and
the number that the logarithm acts on. Logarithm is the inverse operation of
exponentiation (they “undo” eachother). The logarithm of a number x in base
b is the number n such that x = b^{n}. Here is the notation:
This is read “log base b of x equals n.” By definition, if
, then x = b^{n}.
For example:
Consider the equation b ^{x} = n. If we know b and
x, we calculate n where n is a
power of b. If we know x and n, we find b by taking the x^{th} root (or
radical) of
n. If we know b and n, we compute a logarithm to find x. For example:
b^{3} = 27 becomes
so b = 3
4^{x} = 4096 becomes so x = 6
The allowable operations on logarithms are:
• The product rule:
• The quotient rule:
• The power rule:
Using these rules and the definition of the logarithm, we can see how the
logarithm is the inverse of the exponential:
and . To
better understand the inverse relationship between logarithm and exponentia
tion, consider these equations more closely:
The product rule allows me to rewrite
as . By
definition,
produces a number n such that a^{n} = a; clearly, n=1. Therefore,
. Now, consider the other equation:
By definition,
produces the number n such that a^{n} = x. Therefore,
Exercises: Logarithms
1. Calculate
2..Find x.
2.3 The number e and the Natural Logarithm (ln)
The exponential function is a function of base e raised to some power. The
number e ≈ 2.718 is often used to model growth and decay. It is so
important
that it has its own notation:
When the base of an exponential function is not e, it is
possible to change the
base :
This also works for other basechanges.
Logarithms of base e are “natural logarithms.” They also have their own
notation:
Logarithms of base 10 are also commonly used . When the
base is not specified
(log(n)), it can generally be assumed that it is base 10.
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