Supplementary Course Module G: Exponentials and Logarithms
Exponential and Logarithmic Operations
Consider the seemingly innocent enough numerical expression,
3
2
.
There are two different ways of looking at this expression, depending on where we start!
1.
If we start with the
3
, the expression translates to "
3
to the power of
2
" This is an example of a power operation
and can be broken down as follows:
3
{z}
given value
%
2
o
operation
=
9
{z}
answer
That is, given the value
3
and applying the operation of "to the power of
2
", we obtain the answer
9
.
An opposite (or inverse) operation should take an answer and produce the original given value. We would use
operation
q
9
{z}
orig. answer
=
3
{z}
orig. given
3
2
= 9;
p
9 = 3
If we replace the given value of
3
with the variable
x
, we have that
x
2
(power of
2
) and
p
x
(square root) are
opposite operations.
In general, if
n
is a nonzero positive integer,
x
n
(the
n
th
power) and
n
p
x
(the
n
th
root) are opposites.
This is probably the more common way of looking at the expression
3
2
. But what if we start at the
2
?
2. If we start with the
2
, the expression translates to "
2
applied to the base of
3
" and the direction of the operation
arrow changes direction:
2
g
given value
3
.
o
operation
=
9
{z}
answer
That is, given the value
2
and applying the operation of "to the base of
3
", we obtain the answer
9
the exponential operation.
Looking at the opposite operation, this time we need to do something to the
9
in order to obtain the original
given value of
2
. We know the base of the exponential operation was
3
, thus, we apply a logarithmic operation,
3
":
log
3
{z}
operation
9
{z}
orig.
answer
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 Fall '10
 HU
 Exponential Function, Exponentiation, Natural logarithm, Logarithm

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