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 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 (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
exponentiation (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 x^th 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

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 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
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.