Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "Logarithm"

(→‎Problems: added classic)
m (spelling, plurality in links)
Line 1: Line 1:
 
== Introduction ==
 
== 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 <math>\displaystyle x=\log_a{b}</math>.
+
'''Logarithms''' and [[exponents]] are very closely related.  In fact, they are [[inverse function | 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 <math>\displaystyle x=\log_a{b}</math>.
  
 
We would read this as "the logarithm of b, base a, is x".  For example, we know that <math>3^3=27</math>. To express this in Logarithmic notation, we would write it as <math>\log_3 27=3</math>.
 
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>.
Line 22: Line 22:
 
*<math>\displaystyle \log_{a^n} b^n=\log_a b</math>
 
*<math>\displaystyle \log_{a^n} b^n=\log_a b</math>
  
−
Try proving all of these as excercises.
+
Try proving all of these as exercises.
  
 
== Problems ==
 
== Problems ==

Revision as of 10:34, 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 $a^x = b$, then taking the logarithm with base $a$ on both sides will give us $\displaystyle x=\log_a{b}$.

We would read this as "the logarithm of b, base a, is x". For example, we know that $3^3=27$. To express this in Logarithmic notation, we would write it as $\log_3 27=3$.

When a logarithm has no base, it is assumed to be base 10. Thus, $\log(100)$ means $\log_{10}(100)=2$.

Logarithmic Properties

We can use the properties of exponents to build a set of properties for logarithms.

We know that $a^x\cdot a^y=a^{x+y}$. We let $a^x=b$ and $a^y=c$. This also makes $\displaystyle a^{x+y}=bc$. From $a^x = b$ we have $x = \log_a{b}$ and from $a^y=c$ we have $y=\log_a{c}$. So $x+y = \log_a{b}+\log_a{c}$. But we also have from $\displaystyle a^{x+y} = bc$ that $x+y = \log_a{bc}$ Thus, we have found two expressions for $x+y$ establishing the identity:

$\log_a{b} + \log_a{c} = \log_a{bc}.$

Using the laws of exponents, we can derive and prove the following identities:

  • $\log_a b^n=n\log_a b$
  • $\log_a b+ \log_a c=\log_a bc$
  • $\log_a b-\log_a c=\log_a \frac{b}{c}$
  • $(\log_a b)(\log_c d)= (\log_a d)(\log_c b)$
  • $\frac{\log_a b}{\log_a c}=\log_c b$
  • $\displaystyle \log_{a^n} b^n=\log_a b$

Try proving all of these as exercises.

Problems

  1. Evaluate $(\log_{50}{2.5})(\log_{2.5}e)(\ln{2500})$.
  1. Evaluate $(\log_2 3)(\log_3 4)(\log_4 5)\cdots(\log_{2005} 2006)$.
  1. Simplify $\displaystyle \frac 1{\log_2 N}+\frac 1{\log_3 N}+\frac 1{\log_4 N}+\cdots+ \frac 1{\log_{100}}{N}$ where $N=(100!)^3$.
Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=Logarithm&oldid=3890"