Difference between revisions of "Logarithm"
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− | == | + | |
− | + | == Videos On Logarithms== | |
− | + | ||
− | + | [https://youtu.be/fU7B2H4JJCU Introduction to Logarithms] | |
− | = | + | |
− | + | [https://youtu.be/tUUynpSD5DM Logarithms Properties] | |
− | + | ||
− | *<math> | + | ==Powerful use of logarithms== |
− | *<math> | + | |
− | *<math> | + | 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 |
− | *<math>\ | + | 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(5)\approx 1.160964047443681173935159715\ldots</cmath> | ||
+ | <cmath>\log_4(6)\approx 1.292481250360578090726869472\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: | ||
+ | |||
+ | <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=== | ||
+ | |||
+ | 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). | ||
+ | |||
+ | 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 == | ||
+ | |||
+ | # Evaluate <math>(\log_{50}{2.5})(\log_{2.5}e)(\ln{2500})</math>. | ||
+ | # Evaluate <math>(\log_2 3)(\log_3 4)(\log_4 5)\cdots(\log_{2005} 2006)</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>. | ||
+ | |||
+ | == Natural Logarithm == | ||
+ | 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.) | ||
+ | |||
+ | <math>\ln a</math> can also be defined as the area under the curve <math>y=\frac{1}{x}</math> between 1 and a, or <math>\int^a_1 \frac{1}{x}\, dx</math>. | ||
+ | |||
+ | All logarithms are undefined in nonpositive reals, as they are complex. From the identity <math>e^{i\pi}=-1</math>, we have <math>\ln (-1)=i\pi</math>. Additionally, <math>\ln (-n)=\ln n+i\pi</math> for positive real <math>n</math>. | ||
+ | |||
+ | == Problems == | ||
+ | === Introductory === | ||
+ | * 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]] | ||
+ | * 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>. | ||
+ | [[2013_AIME_II_Problems/Problem_2 | Source]] | ||
+ | === Intermediate === | ||
+ | * 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> | ||
+ | [[2006_AIME_I_Problems/Problem_9 | Source]] | ||
+ | |||
+ | * The solutions to the system of equations | ||
+ | <math>\log_{225}x+\log_{64}y=4</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>. | ||
+ | [[2002_AIME_I_Problems/Problem_6 | Source]] | ||
+ | |||
+ | === Olympiad === | ||
+ | |||
+ | ==Video Explanation== | ||
+ | Five-minute Intro to Logarithms w/ examples [https://youtu.be/CxXc_moY-3Q] | ||
+ | |||
+ | ==External Links== | ||
+ | |||
+ | Two-minute Intro to Logarithms [http://www.youtube.com/watch?v=ey7ttABX9SM] | ||
+ | |||
+ | [[Category:Definition]] | ||
+ | [[Category:Functions]] |
Latest revision as of 12:54, 16 October 2023
Contents
Videos On Logarithms
Powerful use of logarithms
Some of the real powerful uses of logarithms come down to never having to deal with massive numbers. ex. : 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.
Therefore by: and identities 1 and 2 above (2 being used twice), we get:
such that:
Discrete Logarithm
An only partially related value is the discrete logarithm, used in cryptography via modular arithmetic. It's the lowest value such that
for given
being integers (as well as
the unknowns being integers).
It's related to the usual logarithm by the fact that if isn't an integer power of
then
is a lower bound on
.
Problems
- Evaluate
.
- Evaluate
.
- Simplify
where
.
Natural Logarithm
The natural logarithm is the logarithm with base e. It is usually denoted , an abbreviation of the French logarithme normal, so that
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
and the symbol
is not used at all. (This is an example of conflicting mathematical conventions.)
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
.
Problems
Introductory
- What is the value of
for which
?
- Positive integers
and
satisfy the condition
Find the sum of all possible values of
.
Intermediate
- The sequence
is geometric with
and common ratio
where
and
are positive integers. Given that
find the number of possible ordered pairs
- The solutions to the system of equations
are
and
. Find
.
Source
Olympiad
Video Explanation
Five-minute Intro to Logarithms w/ examples [1]
External Links
Two-minute Intro to Logarithms [2]