Difference between revisions of "Logarithm"
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All logarithms are undefined in nonnegative reals, as they are complex. From the identity <math>e^{i\pi}=-1</math>, we have <math>\ln (-1)=i\pi</math>. Additionally, <math>\ln (-n)=\ln n+i\pi</math> for positive real <math>n</math>. | All logarithms are undefined in nonnegative reals, as they are complex. From the identity <math>e^{i\pi}=-1</math>, we have <math>\ln (-1)=i\pi</math>. Additionally, <math>\ln (-n)=\ln n+i\pi</math> for positive real <math>n</math>. | ||
+ | |||
+ | == Negative Logarithm == | ||
+ | We define <math>\log_b (-x)</math> to be <math>\log_b(x)+i\log_b(e^\pi)</math> for positive real <math>x</math>. | ||
== Intermediate == | == Intermediate == | ||
=== Example Problem === | === Example Problem === | ||
* [[2006_AIME_I_Problems/Problem_9 | 2006 AIME I Problem 9]] | * [[2006_AIME_I_Problems/Problem_9 | 2006 AIME I Problem 9]] |
Revision as of 21:17, 4 November 2006
Logarithms and exponents are very closely related. In fact, they are inverse functions. This means that logarithms can be used to reverse the result of exponentiation and vice versa, just as addition can be used to reverse the result of subtraction. Thus, if we have , then taking the logarithm with base
on both sides will give us
.
We would read this as "the logarithm of b, base a, is x". For example, we know that . To express the same fact in logarithmic notation we would write
.
Depending on the field, the symbol without a base can have different meanings. Typically, in mathematics through the level of calculus, the symbol is used to refer to a base 10 logarithm. Thus,
means
. Usually, the symbol
(an abbreviation of the French "logarithme normal," meaning "natural logarithm") is introduced to refer to the logarithm base e. However, in higher mathematics such as complex analysis, the base 10 logarithm is typically disposed with entirely, the symbol
is taken to mean the logarithm base e and the symbol
is not used at all. (This is an example of conflicting mathematical conventions.)
Contents
Logarithmic Properties
We can use the properties of exponents to build a set of properties for logarithms.
We know that . We let
and
. This also makes
. From
, we have
, and from
, we have
. So,
. But we also have from
that
. Thus, we have found two expressions for
establishing the identity:
![$\log_a{b} + \log_a{c} = \log_a{bc}.$](http://latex.artofproblemsolving.com/e/b/3/eb36932b21a6a980e7139766aff531b438dad9f1.png)
Using the laws of exponents, we can derive and prove the following identities:
(Known as the product property.)
Try proving all of these as exercises.
Here are some other less useful log properties that follow from these previous ones.
(Known as the inverse property)
(Known as the chain rule.)
Problems
- Evaluate
.
- Evaluate
.
- Simplify
where
.
Natural Logarithm
The natural logarithm of is
. The function
is the inverse of
.
can also be defined as the area under the curve
between 1 and a, or
.
All logarithms are undefined in nonnegative reals, as they are complex. From the identity , we have
. Additionally,
for positive real
.
Negative Logarithm
We define to be
for positive real
.