Difference between revisions of "Logarithm"
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− | '''Logarithms''' and [[exponents]] are very closely related. In fact, they are [[Function/Introduction#The_Inverse_of_a_Function|inverse]] [[function]]s. 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 <math> a^x = b </math>, then taking the logarithm with base <math> a</math> on both sides will give us <math> | + | {{WotWAnnounce|week=Nov 29-Dec 5}} |
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+ | '''Logarithms''' and [[exponents]] are very closely related. In fact, they are [[Function/Introduction#The_Inverse_of_a_Function|inverse]] [[function]]s. 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 <math> a^x = b </math>, then taking the logarithm with base <math> a</math> on both sides will give us <math>x=\log_a{b}</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>. | 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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We can use the properties of exponents to build a set of properties for logarithms. | We can use the properties of exponents to build a set of properties for logarithms. | ||
− | We know that <math>a^x\cdot a^y=a^{x+y}</math>. We let <math> a^x=b</math> and <math> a^y=c </math>. This also makes <math> | + | We know that <math>a^x\cdot a^y=a^{x+y}</math>. We let <math> a^x=b</math> and <math> a^y=c </math>. This also makes <math>a^{x+y}=bc </math>. From <math> a^x = b</math>, we have <math> x = \log_a{b}</math>, and from <math> a^y=c </math>, we have <math> y=\log_a{c} </math>. So, <math> x+y = \log_a{b}+\log_a{c}</math>. But we also have from <math>a^{x+y} = bc</math> that <math> x+y = \log_a{bc}</math>. Thus, we have found two expressions for <math> x+y</math> establishing the identity: |
<center><math> \log_a{b} + \log_a{c} = \log_a{bc}.</math></center> | <center><math> \log_a{b} + \log_a{c} = \log_a{bc}.</math></center> | ||
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*<math>(\log_a b)(\log_c d)= (\log_a d)(\log_c b)</math> | *<math>(\log_a b)(\log_c d)= (\log_a d)(\log_c b)</math> | ||
*<math>\frac{\log_a b}{\log_a c}=\log_c b</math> | *<math>\frac{\log_a b}{\log_a c}=\log_c b</math> | ||
− | *<math> | + | *<math>\log_{a^n} b^n=\log_a b</math> |
Try proving all of these as exercises. | Try proving all of these as exercises. | ||
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# Evaluate <math>(\log_2 3)(\log_3 4)(\log_4 5)\cdots(\log_{2005} 2006)</math>. | # Evaluate <math>(\log_2 3)(\log_3 4)(\log_4 5)\cdots(\log_{2005} 2006)</math>. | ||
− | # Simplify <math> | + | # Simplify <math>\frac 1{\log_2 N}+\frac 1{\log_3 N}+\frac 1{\log_4 N}+\cdots+ \frac 1{\log_{100}N} </math> where <math> N=(100!)^3</math>. |
Revision as of 11:10, 29 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 .
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 .