Difference between revisions of "Logarithm"
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| − | + | == Introduction == | |
| − | + | '''Logarithms''' and [[exponents]] are very closely related. In fact, they are [[inverse functions]]. Basically, 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 | |
| − | + | ||
| − | + | <center><math>\displaystyle x=\log_a{b}.</math></center> | |
| − | When a logarithm has no base, it is assumed to be base 10. | + | |
| + | We would read this as "the logarithm of b base a is x". For example, we know that <math>3^3=27</math>. To express this in [[Logarithmic notation]], we would write it as <math>\log_3 27=3</math>. | ||
| + | |||
| + | When a logarithm has no base, it is assumed to be base 10. Thus, <math>\log(100)</math> means <math>\log_{10}(100)=2</math>. | ||
| + | |||
==Logarithmic Properties== | ==Logarithmic Properties== | ||
| − | + | 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>\displaystyle 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>\displaystyle 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> | ||
| + | |||
| + | Using the laws of exponents, we can derive and prove the following identities: | ||
| + | |||
*<math>\log_a b^n=n\log_a b</math> | *<math>\log_a b^n=n\log_a b</math> | ||
*<math>\log_a b+ \log_a c=\log_a bc</math> | *<math>\log_a b+ \log_a c=\log_a bc</math> | ||
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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>\displaystyle \log_{a^n} b^n=\log_a b</math> |
| + | |||
| + | Try proving all of these as excercises. | ||
| + | |||
| + | == Problems == | ||
| + | |||
| + | # Evaluate <math>(\log_{50}{2.5})(\log_{2.5}e)(\ln{2500}).</math> | ||
| + | # Simplify <math>\displaystyle \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 10:37, 22 June 2006
Introduction
Logarithms and exponents are very closely related. In fact, they are inverse functions. Basically, 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 this in Logarithmic notation, we would write it as 
.
When a logarithm has no base, it is assumed to be base 10. Thus, 
means 
.
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:
Try proving all of these as excercises.
Problems
- Evaluate
- Simplify
where 
.
where 
.
