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Difference between revisions of "Logarithm"

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'''Logarithms''' are right inverses of powers.
  
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== Definition ==
  
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== Videos On Logarithms==
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Logarithms, the [[right inverse]]s of powers can be written as <math>\log_a(b)</math> for some <math>a</math> and <math>b</math>, where <math>a^{\log_a(b)}=b</math>.
  
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[https://youtu.be/fU7B2H4JJCU Introduction to Logarithms]
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== Applications ==
  
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[https://youtu.be/tUUynpSD5DM Logarithms Properties]
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Some of the real powerful uses of logarithms come down to never having to deal with massive numbers. For example,
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<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  
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[https://www.youtube.com/watch?v=Z5myJ8dg_rM Logarithms by Khan Academy]
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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)). For example,
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==Powerful use of logarithms==
 
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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 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>
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<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>
  
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=== Discrete Logarithm ===
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===Discrete Logarithm===
 
  
 
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).  
 
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).  
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== Natural Logarithm ==
 
== Natural Logarithm ==
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The '''natural logarithm''' is the logarithm with base [[e]].  It is usually denoted <math>\ln</math>, an abbreviation of the French ''logarithme normal'', so that <math> \ln(x) = \log_e(x).</math>  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 <math>e</math> and the symbol <math>\ln</math> is not used at all.  (This is an example of conflicting [[mathematical convention]]s.)
 
The '''natural logarithm''' is the logarithm with base [[e]].  It is usually denoted <math>\ln</math>, an abbreviation of the French ''logarithme normal'', so that <math> \ln(x) = \log_e(x).</math>  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 <math>e</math> and the symbol <math>\ln</math> is not used at all.  (This is an example of conflicting [[mathematical convention]]s.)
  
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== Problems ==
 
== Problems ==
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=== Introductory ===
 
=== Introductory ===
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* What is the value of <math>a</math> for which <math>\frac1{\log_2a}+\frac1{\log_3a}+\frac1{\log_4a}=1</math>?
 
* What is the value of <math>a</math> for which <math>\frac1{\log_2a}+\frac1{\log_3a}+\frac1{\log_4a}=1</math>?
 
[[2015_AMC_12A_Problems/Problem_14 | Source]]
 
[[2015_AMC_12A_Problems/Problem_14 | Source]]
 
* Positive integers <math>a</math> and <math>b</math> satisfy the condition <math>\log_2(\log_{2^a}(\log_{2^b}(2^{1000})))=0.</math> Find the sum of all possible values of <math>a+b</math>.  
 
* Positive integers <math>a</math> and <math>b</math> satisfy the condition <math>\log_2(\log_{2^a}(\log_{2^b}(2^{1000})))=0.</math> Find the sum of all possible values of <math>a+b</math>.  
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[[2013_AIME_II_Problems/Problem_2 | Source]]
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[[2013_AIME_II_Problems/Problem_2|Source]]
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=== Intermediate ===
 
=== Intermediate ===
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* 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>
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[[2006_AIME_I_Problems/Problem_9 | Source]]
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* 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>
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[[2006_AIME_I_Problems/Problem_9|Source]]
  
 
* The solutions to the system of equations
 
* The solutions to the system of equations
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<math>\log_{x}225-\log_{y}64=1</math>
 
<math>\log_{x}225-\log_{y}64=1</math>
 
are <math>(x_1,y_1)</math> and <math>(x_2,y_2)</math>. Find <math>\log_{30}\left(x_1y_1x_2y_2\right)</math>.
 
are <math>(x_1,y_1)</math> and <math>(x_2,y_2)</math>. Find <math>\log_{30}\left(x_1y_1x_2y_2\right)</math>.
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[[2002_AIME_I_Problems/Problem_6 | Source]]
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[[2002_AIME_I_Problems/Problem_6|Source]]
  
 
=== Olympiad ===
 
=== Olympiad ===
  
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==Video Explanation==
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{{problem}}
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Five-minute Intro to Logarithms w/ examples [https://youtu.be/CxXc_moY-3Q]
 
  
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==External Links==
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== Videos ==
  
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Two-minute Intro to Logarithms [http://www.youtube.com/watch?v=ey7ttABX9SM]
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* [//youtu.be/CxXc_moY-3Q Five-minute Intro to Logarithms]
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* [//youtu.be/fU7B2H4JJCU Introduction to Logarithms]
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* [//youtu.be/tUUynpSD5DM Logarithms Properties]
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* [//www.youtube.com/watch?v=Z5myJ8dg_rM Khan Academy]
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* [//www.youtube.com/watch?v=ey7ttABX9SM Two-minute Intro to Logarithms]
  
 
[[Category:Definition]]
 
[[Category:Definition]]
 
[[Category:Functions]]
 
[[Category:Functions]]
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{{stub}}

Revision as of 20:44, 9 March 2025

Logarithms are right inverses of powers.

Definition

Logarithms, the right inverses of powers can be written as $\log_a(b)$ for some $a$ and $b$, where $a^{\log_a(b)}=b$.

Applications

Some of the real powerful uses of logarithms come down to never having to deal with massive numbers. For example, \[((((((3^5)^6)^7)^8)^9)^{10})^{11}=\underbrace{1177\ldots 1}_{\text{793549 digits}}\] 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)). For example,

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

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:

\[(\log_{50}{2.5})(\log_{2.5}e)(\ln{2500})\]

  1. Evaluate:

\[(\log_2 3)(\log_3 4)(\log_4 5) \cdots (\log_{2005} 2006)\]

  1. Simplify the following expression:

\[\frac{1}{\log_2 N} + \frac{1}{\log_3 N} + \frac{1}{\log_4 N} + \cdots + \frac{1}{\log_{100} N},\]

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

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Videos

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