# 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 base-changes.

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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