Difference between revisions of "Logarithm"
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==Conventions== | ==Conventions== | ||
− | Depending on the field, the symbol <math>\log</math> without a base can have different meanings. | + | Depending on the field, the symbol <math>\log</math> without a base can have different meanings. Sometimes in high schools, the symbol is used to refer to a base 10 logarithm. Thus, <math>\log(100)</math> can mean <math>\log_{10}(100)=2</math>. In these contexts, the symbol <math>\ln</math> (an abbreviation of the French "logarithme normal," meaning "natural logarithm") is introduced to refer to the logarithm base [[e]], or '''natural logarithm'''. However, the choice of base 10 is arbitrary, and convenient only for computations in a base-10 number system. The natural logarithm, however, has many convenient mathematical properties, so practicing mathematicians often take the symbol <math>\log</math> is taken to mean the natural logarithm and do not use the symbol <math>\ln</math>. (This is an example of conflicting [[mathematical convention]]s.) In addition, the notation <math>\lg</math> is often used by [[combinatorics | combinatorists]] and computer scientists to refer to the logarithm base <math>2</math>. Occasionally, the base of the logarithms is irrelevant. |
==Logarithmic Properties== | ==Logarithmic Properties== |
Revision as of 14:18, 29 March 2009
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
.
Contents
Conventions
Depending on the field, the symbol without a base can have different meanings. Sometimes in high schools, the symbol is used to refer to a base 10 logarithm. Thus,
can mean
. In these contexts, the symbol
(an abbreviation of the French "logarithme normal," meaning "natural logarithm") is introduced to refer to the logarithm base e, or natural logarithm. However, the choice of base 10 is arbitrary, and convenient only for computations in a base-10 number system. The natural logarithm, however, has many convenient mathematical properties, so practicing mathematicians often take the symbol
is taken to mean the natural logarithm and do not use the symbol
. (This is an example of conflicting mathematical conventions.) In addition, the notation
is often used by combinatorists and computer scientists to refer to the logarithm base
. Occasionally, the base of the logarithms is irrelevant.
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 nonpositive reals, as they are complex. From the identity , we have
. Additionally,
for positive real
.
Problems
Introductory
Intermediate
- The sequence
is geometric with
and common ratio
where
and
are positive integers. Given that
find the number of possible ordered pairs