Difference between revisions of "Logarithm"
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We would read this as "the logarithm of b, base a, is x". For example, we know that <math>3^4=81</math>. To express the same fact in logarithmic notation we would write <math>\log_3 81=4</math>. | We would read this as "the logarithm of b, base a, is x". For example, we know that <math>3^4=81</math>. To express the same fact in logarithmic notation we would write <math>\log_3 81=4</math>. | ||
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+ | ==Conventions== | ||
Depending on the field, the symbol <math>\log</math> 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, <math>\log(100)</math> means <math>\log_{10}(100)=2</math>. Usually, the symbol <math>\ln</math> (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 <math>\log</math> is taken to mean the logarithm base e and the symbol <math>\ln</math> is not used at all. (This is an example of conflicting [[mathematical convention]]s.) | Depending on the field, the symbol <math>\log</math> 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, <math>\log(100)</math> means <math>\log_{10}(100)=2</math>. Usually, the symbol <math>\ln</math> (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 <math>\log</math> is taken to mean the logarithm base e and the symbol <math>\ln</math> is not used at all. (This is an example of conflicting [[mathematical convention]]s.) | ||
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All logarithms are undefined in nonpositive 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 nonpositive 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>. | ||
− | == | + | == Problems == |
− | === | + | === Introductory === |
− | * [[2006_AIME_I_Problems/Problem_9 | | + | === Intermediate === |
+ | * The [[sequence]] <math> a_1, a_2, \ldots </math> is [[geometric sequence|geometric]] with <math> a_1=a </math> and common [[ratio]] <math> r, </math> where <math> a </math> and <math> r </math> are positive integers. Given that <math> \log_8 a_1+\log_8 a_2+\cdots+\log_8 a_{12} = 2006, </math> find the number of possible ordered pairs <math> (a,r). </math> | ||
+ | [[2006_AIME_I_Problems/Problem_9 | Source]] | ||
+ | === Olympiad === | ||
+ | |||
+ | |||
+ | [[Category:Definition]] | ||
+ | [[Category:Function]] |
Revision as of 19:53, 30 November 2007
This is an AoPSWiki Word of the Week for Nov 29-Dec 5 |
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. 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.)
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:
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