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"

(→‎Intermediate)
m (Fixed punctuation errors, capitalization, its/it's, spaces before colons, a/an.)
 
(3 intermediate revisions by one other user not shown)
Line 9: Line 9:
 
==Powerful use of logarithms==
 
==Powerful use of logarithms==
  
−
Some of the real powerful uses of logarithms, come down to never having to deal with massive numbers. ex. :<cmath>((((((3^5)^6)^7)^8)^9)^{10})^{11}=\underbrace{1177\ldots 1}_{\text{793549 digits}}</cmath> would be a pain to have to calculate any time you wanted to use it (say in a comparison of large numbers). its natural logarithm though (partly due to  
+
Some of the real powerful uses of logarithms come down to never having to deal with massive numbers. ex. :<cmath>((((((3^5)^6)^7)^8)^9)^{10})^{11}=\underbrace{1177\ldots 1}_{\text{793549 digits}}</cmath> would be a pain to have to calculate any time you wanted to use it (say in a comparison of large numbers). Its natural logarithm though (partly due to  
−
left to right parenthesized exponentiation) is only 7 digits before the decimal point. Comparing the logs of the numbers to a given precision can allow easier comparision than computing and comparing the numbers themselves. Logs also allow (with repetition) to turn left to right exponentiation into power towers (especially useful for tetration (exponentiation repetition with the same exponent)). ex.
+
left to right parenthesized exponentiation) is only 7 digits before the decimal point. Comparing the logs of the numbers to a given precision can allow easier comparison than computing and comparing the numbers themselves. Logs also allow (with repetition) to turn left to right exponentiation into power towers (especially useful for tetration (exponentiation repetition with the same exponent)). ex.
  
 
<cmath>\log_4(3)\approx 0.7924812503605780907268694720\ldots</cmath>
 
<cmath>\log_4(3)\approx 0.7924812503605780907268694720\ldots</cmath>
Line 17: Line 17:
 
<cmath>\log_4(\log_4(3))\approx -0.1677756462730553083259853611\ldots</cmath>
 
<cmath>\log_4(\log_4(3))\approx -0.1677756462730553083259853611\ldots</cmath>
  
−
Therefore by : <cmath>(a^b)^c=a^{bc}</cmath> and identities 1 and 2 above ( 2 being used twice) we get:
+
Therefore by: <cmath>(a^b)^c=a^{bc}</cmath> and identities 1 and 2 above (2 being used twice), we get:
  
−
<cmath>\log_4(\log_4(3))+(\log_4(5)+\log_4(6))\approx 2.285669651531203956336043826\ldots=x</cmath> such that :<cmath>(^24)^x=4^{4^x}\approx(3^5)^6</cmath>
+
<cmath>\log_4(\log_4(3))+(\log_4(5)+\log_4(6))\approx 2.285669651531203956336043826\ldots=x</cmath> such that:<cmath>(^24)^x=4^{4^x}\approx(3^5)^6</cmath>
  
  
 
===Discrete Logarithm===
 
===Discrete Logarithm===
  
−
A only partially related value is the discrete logarithm, used in [[cryptography]] via [[modular arithmetic]]. It's the lowest value <math>c</math> such that, <math>a^c=mx+b</math> for given <math>a,m,b</math> being integers (as well as <math>c,x</math> the unknowns being integer).  
+
An only partially related value is the discrete logarithm, used in [[cryptography]] via [[modular arithmetic]]. It's the lowest value <math>c</math> such that <math>a^c=mx+b</math> for given <math>a,m,b</math> being integers (as well as <math>c,x</math> the unknowns being integers).  
  
−
Its related to the usual logarithm, by the fact that if <math>b</math> isn't an integer power of <math>a</math> then <math>\lceil \log_a(m)\rceil</math> is a lower bound on <math>c</math>
+
It's related to the usual logarithm by the fact that if <math>b</math> isn't an integer power of <math>a</math> then <math>\lceil \log_a(m)\rceil</math> is a lower bound on <math>c</math>.
  
 
== Problems ==
 
== Problems ==
Line 59: Line 59:
  
 
=== Olympiad ===
 
=== Olympiad ===
 +
 +
==Video Explanation==
 +
Five-minute Intro to Logarithms w/ examples [https://youtu.be/CxXc_moY-3Q]
  
 
==External Links==
 
==External Links==
 +
 
Two-minute Intro to Logarithms [http://www.youtube.com/watch?v=ey7ttABX9SM]
 
Two-minute Intro to Logarithms [http://www.youtube.com/watch?v=ey7ttABX9SM]
  
 
[[Category:Definition]]
 
[[Category:Definition]]
 
[[Category:Functions]]
 
[[Category:Functions]]

Latest revision as of 11:54, 16 October 2023


Videos On Logarithms

Introduction to Logarithms

Logarithms Properties

Powerful use of logarithms

Some of the real powerful uses of logarithms come down to never having to deal with massive numbers. ex. :\[((((((3^5)^6)^7)^8)^9)^{10})^{11}=\underbrace{1177\ldots 1}_{\text{793549 digits}}\] would be a pain to have to calculate any time you wanted to use it (say in a comparison of large numbers). Its natural logarithm though (partly due to left to right parenthesized exponentiation) is only 7 digits before the decimal point. Comparing the logs of the numbers to a given precision can allow easier comparison than computing and comparing the numbers themselves. Logs also allow (with repetition) to turn left to right exponentiation into power towers (especially useful for tetration (exponentiation repetition with the same exponent)). ex.

\[\log_4(3)\approx 0.7924812503605780907268694720\ldots\] \[\log_4(5)\approx 1.160964047443681173935159715\ldots\] \[\log_4(6)\approx 1.292481250360578090726869472\ldots\] \[\log_4(\log_4(3))\approx -0.1677756462730553083259853611\ldots\]

Therefore by: \[(a^b)^c=a^{bc}\] and identities 1 and 2 above (2 being used twice), we get:

\[\log_4(\log_4(3))+(\log_4(5)+\log_4(6))\approx 2.285669651531203956336043826\ldots=x\] such that:\[(^24)^x=4^{4^x}\approx(3^5)^6\]


Discrete Logarithm

An only partially related value is the discrete logarithm, used in cryptography via modular arithmetic. It's the lowest value $c$ such that $a^c=mx+b$ for given $a,m,b$ being integers (as well as $c,x$ the unknowns being integers).

It's related to the usual logarithm by the fact that if $b$ isn't an integer power of $a$ then $\lceil \log_a(m)\rceil$ is a lower bound on $c$.

Problems

  1. Evaluate $(\log_{50}{2.5})(\log_{2.5}e)(\ln{2500})$.
  2. Evaluate $(\log_2 3)(\log_3 4)(\log_4 5)\cdots(\log_{2005} 2006)$.
  3. Simplify $\frac 1{\log_2 N}+\frac 1{\log_3 N}+\frac 1{\log_4 N}+\cdots+ \frac 1{\log_{100}N}$ where $N=(100!)^3$.

Natural Logarithm

The natural logarithm is the logarithm with base e. It is usually denoted $\ln$, an abbreviation of the French logarithme normal, so that $\ln(x) = \log_e(x).$ However, in higher mathematics such as complex analysis, the base 10 logarithm is typically disposed with entirely, the symbol $\log$ is taken to mean the logarithm base $e$ and the symbol $\ln$ is not used at all. (This is an example of conflicting mathematical conventions.)

$\ln a$ can also be defined as the area under the curve $y=\frac{1}{x}$ between 1 and a, or $\int^a_1 \frac{1}{x}\, dx$.

All logarithms are undefined in nonpositive reals, as they are complex. From the identity $e^{i\pi}=-1$, we have $\ln (-1)=i\pi$. Additionally, $\ln (-n)=\ln n+i\pi$ for positive real $n$.

Problems

Introductory

  • What is the value of $a$ for which $\frac1{\log_2a}+\frac1{\log_3a}+\frac1{\log_4a}=1$?

Source

  • Positive integers $a$ and $b$ satisfy the condition $\log_2(\log_{2^a}(\log_{2^b}(2^{1000})))=0.$ Find the sum of all possible values of $a+b$.

Source

Intermediate

  • The sequence $a_1, a_2, \ldots$ is geometric with $a_1=a$ and common ratio $r,$ where $a$ and $r$ are positive integers. Given that $\log_8 a_1+\log_8 a_2+\cdots+\log_8 a_{12} = 2006,$ find the number of possible ordered pairs $(a,r).$

Source

  • The solutions to the system of equations

$\log_{225}x+\log_{64}y=4$

$\log_{x}225-\log_{y}64=1$ are $(x_1,y_1)$ and $(x_2,y_2)$. Find $\log_{30}\left(x_1y_1x_2y_2\right)$. Source

Olympiad

Video Explanation

Five-minute Intro to Logarithms w/ examples [1]

External Links

Two-minute Intro to Logarithms [2]

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=Logarithm&oldid=199740"