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 an = x, a is the base, n is the exponent,
and x is the nth power of a. This is equivalent to raising a to the nth power.
For example:

Any number raised to the 1st power is itself and any number raised to the 0th
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 :

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 (x2 + 3x) + (2x2 + 2x + 5)

2. (2x2)(x2)

3. Calculate (32)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
(they “undo” eachother). The logarithm of a number x in base
b is the number n such that x = bn. Here is the notation:

This is read “log base b of x equals n.” By definition, if , then x = bn.
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 xth root (or radical) of
n. If we know b and n, we compute a logarithm to find x. For example:

b3 = 27 becomes so b = 3

4x = 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 an = a; clearly, n=1. Therefore,
. Now, consider the other equation:

By definition, produces the number n such that an = 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

This also works for other base-changes.

Logarithms of base e are “natural logarithms.” They also have their own

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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