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SEO Keyword summary for en.wikipedia.org/wiki/cologarithm
Keywords are extracted from the main content of your website and are the primary indicator of the words this page could rank for. By frequenty count we expect your focus keyword to be log
Focus keyword
Short and long tail
Short Tail Keywords log logarithm displaystyle |
long Tail Keywords (2 words) new york natural logarithm university press real number cambridge university |
long Tail Keywords (3 words) cambridge university press berlin new york new york springer-verlag x and y positive real number york john wiley new york john |
en.wikipedia.org On-Page SEO Scan
Descriptive Elements
The <head> element of a en.wikipedia.org/wiki/cologarithm page is used to inform the browser and visitors of the page about the general meta information. The head section of the page is where we place the page title, the definition of the HTML version used, the language of in which the page is written. In the head section we can also include JavaScript and CSS (markup) files for the page.
Page title
Title length
logarithm wikipedia
Meta description
Meta description legth
Meta description SEO
No meta relevance in the description detected !
Content SEO
Number of Words
Spam detected?
Headings
Heading distribution
Heading normalisation
Heading SEO impact
Emphasis (bold and italic)
Emphasis SEO impact
Images
Number of images
Images dimensions
Image alt descriptions
Images SEO impact
wikipedia free encyclopedia featured article displaystyle scriptstyle leftbeginmatrixscriptstyle texttermtexttermscriptstyle textsummandtextsummandscriptstyle textaddendtextaddendscriptstyle textaugendtextaddendendmatrixright textsum textminuendtextsubtrahendendmatrixright textdifference textfactortimes textfactorscriptstyle textmultipliertimes textmultiplicandendmatrixright textproduct frac textdividendscriptstyle textdivisor exscriptstyle textnumeratorscriptstyle textdenominatorendmatrixright textfractionscriptstyle textquotientscriptstyle textratioendmatrixright textbasetextexponentscriptstyle textbasetextpowerendmatrixright textpower sqrttextdegreescriptstyle textradicand textroot log textbasetextantilogarithm textlogarithm bxylog bxlog graph showing logarithmic curve crossing xaxis approaching minus infinity along yaxis byx ylog xby xmapsto textstyle xblog yblog cdot bfrac xylog bleftxprightplog left right bsqrtpxfrac bxp sqrt bxfrac kxlog xlog exlog bxby tfrac int dyy logcos theta isin itheta beginalignedlog approx endaligned clog cdcd cdleft crightd dlog sqrtdccfrac slide rule two rectangles logarithmically ticked axes arrangement add distance from indicating product mathbb bxy bcolon fxyfxfy graphs functions logarithm function line touching one point ddxlog xln ddxln fxfrac fxfx lnxdxxlnxxc hyperbola part area underneath shaded grey tint xdx beginalignedlntuint tufrac xdxstackrel xdxint ttufrac lntint ufrac wdwlntlnuendaligned depicted twice split into different parts beginalignedlntrint trfrac wrleftrwr dwrightrint wdwrlntendaligned cdots nsum nfrac sum klnn leftx animation increasingly good approximations beginalignedlnzfrac infty kkendaligned ldots zzfrac lnz operatorname artanh leftfrac afrac zexpy lnzylna zfrac lnn lnxapprox mathrm mxrightmln photograph nautilus shell value mark over time its increasing very quickly even scale three asymmetric pdf curves bar chart superimposed second differ slightly but both decrease similar fashion oval shape trajectories particles sksum ipilnpi triangle are removed iterated way four octaves shown linear beginaligned xlnx xint xfrac lntdt lnnln illustration polar form described arrow equivalently length angle eaz rsqrt beginalignedzxiyrcos varphi rcosvarphi kpi isinvarphi density plot middle there black negative axis hue jumps sharply evolves smoothly otherwise eivarphi cos beginalignedzrleftcos rightrleftcosvarphi rightreivarphi elnreivarphi elnrivarphi eakendaligned aklnrivarphi bnx szsum icon kleftblog bxrightlog bxcdot edit wikidata wikimedia foundation powered mediawiki
Mobile SEO en.wikipedia.org/wiki/cologarithm
Mobile rendering
Mobile optimizations
Responsive design detected (mobile css)
No flash detected !
Mobile improvement
Marketing / lead generation for en.wikipedia.org/wiki/cologarithm
Social Media
Facebook shares | Facebook likes | ||
Facebook comments | Tweets | ||
Google +1 |
Conversion form
Search form
Analytics
Online presence
SERP Preview
SERP Title
SERP Link
SERP Description
Domain Level SEO
Domain name
16 characters long
Domain name SEO Impact
Path name
log found in path !
logarithm found in path !
Structured data
Publisher Markup
Other Structured data
Website configuration
Correct processing of non-existing pages?
Favicon icon found?
Robots.txt found?
Sitemap found?
Navigation and internal links
Navigation
Url seperator
Human readable urls
Number of links
Link SEO Impact
statistics
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httpsenwikipediaorgwindexphptitlelogarithmoldid1248847539cologarithm
page information
printable version
cologarithm
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wiki read
arithmetic operations
addition 1
subtraction 1
multiplication 2
product
division 2
fractionquotientratiodisplaystyle scriptstyle leftbeginmatrixscriptstyle textfractionscriptstyle textquotientscriptstyle textratioendmatrixright
exponentiation 3
root extraction 3
mathematics
exponent
common logarithms
natural logarithm
physics
derivative
binary logarithm algorithm
computer science
napier john
navigators
surveying
logarithm table
sum
slide rule
leonhard euler
exponential function
logarithmic scale
decibel
ratio as logarithms
sound pressure
ph
acidity
aqueous solution
formulae
complexity of algorithms
fractals
frequency
intervals
prime numbers
approximating
factorial
psychophysics
forensic accounting
complex logarithm
inverse
discrete logarithm
publickey cryptography
graphs
xaxis
does not meet it
reals
list of logarithmic identities
scientific calculators
irrational
mathematical analysis
decimal
decimal digits
integer
next integer above
information theory
nats
bits
binary form
music theory
octave
cents
semitone
conventional equal temperament
photography
exposure values
light levels
exposure times
apertures
film speeds
international organization for standardization
history of logarithms
function
mirifici logarithmorum canonis descriptio
prosthaphaeresis
jost brgi
archimedes
quadrature
hyperbola
grgoire de saintvincent
the quadrature of the parabola
geometric progression
argument
arithmetic progression
a a de sarasa
christiaan huygens
james gregory
leibniz
integral
roger cotes
encyclopdia britannica
astronomy
celestial navigation
pierresimon laplace
table of logarithms
henry briggs
integer part
fractional part
mantissa
interpolation
strictly increasing
trigonometric identities
trigonometric functions
gunters rule
william oughtred
intermediate value theorem
increasing function
reflecting
diverges to infinity
differentiable function
chain rule
slope
tangent
antiderivative
logarithmic derivative
logarithmic differentiation
related formulas
definite integral
fundamental theorem of calculus
integration by substitution
harmonic series
infinity
converges
eulermascheroni constant
quicksort
algebraic
transcendental
1
almost all
transcendental function
gelfondschneider theorem
ti83 plus
power series
arithmeticgeometric mean
newtons method
cordic
bit shifts
recursively
series
partial sums
converge
limit
taylor series
inverse hyperbolic tangent
sigma notation
rate of convergence
carl friedrich gauss
arithmetic mean
geometric mean
los alamos national laboratory
manhattan project
richard feynman
connection machine
nautilus
scale invariance
logarithmic spiral
benfords law
selfsimilarity
tsiolkovsky rocket equation
fenske equation
nernst equation
goldmark
papiermarks
german hyperinflation in the 1920s
unit of measurement
logarithmicscale
quantities
ratios
power
voltage
acoustics
absorbance
spectrometry
optics
signaltonoise ratio
noise
signal
peak signaltonoise ratio
image compression
moment magnitude scale
richter magnitude scale
apparent magnitude
chemistry
activity
hydronium
hydrogen
ions
moll1
semilog
loglog
power laws
human perception
hicks law
fittss law
weberfechner law
stimulus
sensation
stevenss power law
probability density functions
population of the 237 countries
probability theory
law of large numbers
fair coin
approaches onehalf
law of the iterated logarithm
lognormal distributions
random variable
normal distribution
maximumlikelihood estimation
statistical models
likelihood function
parameter
independent
data sets
logarithm transformation
data transformation
analysis of algorithms
performance
algorithms
divide a problem
binary search algorithm
merge sort
approximately proportional to
uniform cost model
grow logarithmically
natural number
memory
billiards
billiard table
reflections
entropy
statistical thermodynamics
boltzmann constant
entropy in information theory
lyapunov exponents
dynamical system
chaotic
deterministic
equilateral triangles
dimension
sierpinski triangle
hausdorff dimension
counting the number of boxes
equal temperament
pitch
12tone equal temperament
notea
hz
112 tone
just major third
major third
tritone
number theory
prime number theorem
proportional
offset logarithmic integral
riemann hypothesis
conjectures
erdskac theorem
prime factors
stirlings formula
complex numbers
imaginary unit
complex plane
polar form
absolute value
origin
argument
sine
cosine
radian
turns
branches
hue
eulers formula
trigonometric functions
complex exponential
branch cut
multivalued functions
logarithm of a matrix
matrix exponential
padic logarithm
padic exponential
differential geometry
exponential map
tangent space
manifold
neighborhood
finite groups
public key cryptography
diffiehellman key exchange
cryptographic
zechs logarithm
finite field
superlogarithm 4
iterated logarithm
lambert w function
logit
double exponential function
tetration 4
logistic function
group theory
group isomorphism
haar measure
lebesgue measure
semiring
probability semiring
semifield
logsumexp
isomorphism
log semiring
logarithmic oneforms
complex analysis
algebraic geometry
differential forms
poles
polylogarithm
riemann zeta function
decimal exponent
index of logarithm articles
halmos p
stringham i
degree
isbn
oclc
mcgrawhill
springerverlag
goodrich michael t
tamassia roberto
iso 800002
the chicago manual of style
historia mathematica
arxiv
s2cid
robertson edmund f
mactutor history of mathematics archive
university of st andrews
the twoyear college mathematics journal
jstor
florian cajori
american mathematical monthly
abramowitz milton
stegun irene a
handbook of mathematical functions with formulas graphs and mathematical tables
dover publications
the history of mathematical tables from sumer to spreadsheets
oxford university press
princeton university press
devlin keith
lang serge
undergraduate texts in mathematics
wolfram research
kline morris
john wiley sons
nomizu katsumi
baker alan
cambridge university press
issn
kahan w
hillis danny
bibcode
oreilly
pmid
iupac
nadel lynn
citeseerx
tabachnikov serge
american mathematical society
addisonwesley
knuth donald
the art of computer programming
eco umberto
harvard university press
world scientific
crc press
nevanlinna rolf herman
higham nicholas
neukirch jrgen
springerverlag
niederreiter harald
bourbaki nicolas
ambartzumian rv
olver frank w j
nist handbook of mathematical functions
weisstein eric w
mathworld
encyclopedia of mathematics
ems press
chisholm hugh
encyclopdia britannica
hyperoperations
successor 0
pentation 5
hexation 6
predecessor 0
superroot 4
ackermann function
conway chained arrow notation
grzegorczyk hierarchy
knuths uparrow notation
steinhausmoser notation
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PageTitle | 100% | Far too many sites lack a page title. A page title is the first thing that shows in the search results so always use the title element. | |
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Gzip compression | 30% | This site does not use Gzip compression. Pages may not display as fast as they could | |
Keywords in Domainname | 30% | There are no important keywords in your domain name | |
Keywords in domain path | 100% | There are important keywords in the domain path | |
Structured Data | 100% | Structured data makes it easier for search engines to index your website | |
Inline css | 0% | Do not use inline css declarations. Inline css will slow down the rendering of the website. We detected 536 inline style declarations ( <a style="color:green">) with a size of 14328 bytes | |
Excessive use of the same words | 100% | There is no indication that there are one or more keywords that are used excessively. | |
Frames or iframes | 100% | Perfect, detected not (i)frames on your webpagina | |
Flash | 100% | Perfect, we detected no flash objects on your page | |
Css | 30% | We detected too much (2) CSS files on your page. Css files block the loading of a webpage. | |
Javascript | 100% | Perfect, we did not detect too many blocking JavaScript files | |
Mobile Website | 100% | Perfect, we found a responsive design for mobile users | |
Most important heading | 100% | Perfect, we detected a correct use of the most important (h1) heading! | |
Normalized headings | 40% | We dit not font a normalized heading structure. A heading 2 (h2) for example should be followed by a heading of an equal level (h2), a child heading (h3) or even a aprent heading (h1). |
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https://upload.wikimedia.org/wikipedia/commons/thumb/9/9f/binary_logarithm_plot_with_grid.png/300px-binary_logarithm_plot_with_grid.png height: 227 width: 300 description: graph showing a logarithmic curve, crossing the x-axis at x= 1 and approaching minus infinity along the y-axis. |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/32d862a261be92079096455ce1af882eb1c15f99 height: height attribute not set width: width attribute not set description: {\displaystyle b^{y}=x.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/2c3f620a5eabedc8d4791da4f800c50e570212a4 height: height attribute not set width: width attribute not set description: {\displaystyle 2^{3}=8.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/181577dca8c4b040c22f3a8a794a772496c5ba67 height: height attribute not set width: width attribute not set description: {\displaystyle y=\log _{b}x} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/aed854c1c6aed9b31c9d1140d89cacb98c5229dd height: height attribute not set width: width attribute not set description: {\displaystyle x=b^{y}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/7b51b8dcca3e64714e2f3aba12bb8c68812f77cc height: height attribute not set width: width attribute not set description: {\displaystyle b^{y}=x} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/873987f9618fe2c30ce4e72cbd1a967ff759c1d4 height: height attribute not set width: width attribute not set description: {\displaystyle x\mapsto b^{x}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/ac8b94168610cb7ed4c84db6c07c952c86b090a5 height: height attribute not set width: width attribute not set description: {\textstyle \log _{2}\!{\frac {1}{2}}=-1} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/b7133bdd52feadc28a28fe4080716ebd22c813d0 height: height attribute not set width: width attribute not set description: {\textstyle 2^{-1}={\frac {1}{2^{1}}}={\frac {1}{2}}.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/a3f8b602ea0c566a881f48b98516fcd1535adfc4 height: height attribute not set width: width attribute not set description: {\displaystyle x=b^{\,\log _{b}x}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/b45059aa4f1349f9b5fae86c0d31015b3f55e434 height: height attribute not set width: width attribute not set description: {\displaystyle y=b^{\,\log _{b}y}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/3e4f0c2d6cea945fa78f11152fe8127a0649d65a height: height attribute not set width: width attribute not set description: {\textstyle \log _{b}(xy)=\log _{b}x+\log _{b}y} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/5edf603416a8561c82c90c121aef2db207d385fb height: height attribute not set width: width attribute not set description: {\textstyle \log _{3}243=\log _{3}(9\cdot 27)=\log _{3}9+\log _{3}27=2+3=5} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/a87dae414cbad13958321eb275688c81afc52c3f height: height attribute not set width: width attribute not set description: {\textstyle \log _{b}\!{\frac {x}{y}}=\log _{b}x-\log _{b}y} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/cbb8136de3357cf09002709aa2c608431aa819ff height: height attribute not set width: width attribute not set description: {\textstyle \log _{2}16=\log _{2}\!{\frac {64}{4}}=\log _{2}64-\log _{2}4=6-2=4} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/02c0ee92ecd37daeb1e8e8375a09fc073f61b529 height: height attribute not set width: width attribute not set description: {\textstyle \log _{b}\left(x^{p}\right)=p\log _{b}x} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/9999f9b43799c609893876c509df0d2c8f6c752f height: height attribute not set width: width attribute not set description: {\textstyle \log _{2}64=\log _{2}\left(2^{6}\right)=6\log _{2}2=6} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/68ca3b6cc8ff1c0192fb0e9206d32b14aec60e02 height: height attribute not set width: width attribute not set description: {\textstyle \log _{b}{\sqrt[{p}]{x}}={\frac {\log _{b}x}{p}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/5c3e999db5ec9dbc719770862d56466dcf2b4421 height: height attribute not set width: width attribute not set description: {\textstyle \log _{10}{\sqrt {1000}}={\frac {1}{2}}\log _{10}1000={\frac {3}{2}}=1.5} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/6575863d29b357a7ba1ce0a4c1bb93a53b19e671 height: height attribute not set width: width attribute not set description: {\displaystyle \log _{b}x={\frac {\log _{k}x}{\log _{k}b}}.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/b69f8613162e2ae7c46c99bc72402644e16ef317 height: height attribute not set width: width attribute not set description: {\displaystyle \log _{b}x={\frac {\log _{10}x}{\log _{10}b}}={\frac {\log _{e}x}{\log _{e}b}}.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/c3fd9660ce8c1b5ba4ef5d2f1f08a8ed5dde5a08 height: height attribute not set width: width attribute not set description: {\displaystyle b=x^{\frac {1}{y}},} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/2311f2504e15b12a0209f669134985b6b7deac7e height: height attribute not set width: width attribute not set description: {\displaystyle x=b^{\,\log _{b}x}=b^{y}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/88ffe0bbb43942183114c150f227a11184b4c11e height: height attribute not set width: width attribute not set description: {\displaystyle {\tfrac {1}{y}}.} |
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https://upload.wikimedia.org/wikipedia/commons/thumb/a/ae/log4.svg/260px-log4.svg.png height: 226 width: 260 description: no alt description found |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/dfa604c3444753377b366fd0e105138d408338f8 height: height attribute not set width: width attribute not set description: {\displaystyle \log _{10}\,(\,10\,x\,)\ =\;\log _{10}10\ +\;\log _{10}x\ =\ 1\,+\,\log _{10}x\,.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/520963f5a439ba8538bb12cfe16fd0345e641385 height: height attribute not set width: width attribute not set description: {\textstyle \int {\frac {dy}{y}}.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/dea630241d3f7d3497fb2fcae4dacb2434cc2010 height: height attribute not set width: width attribute not set description: {\displaystyle \log(\cos \theta +i\sin \theta )=i\theta .} |
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https://upload.wikimedia.org/wikipedia/commons/thumb/8/82/logarithms_britannica_1797.png/360px-logarithms_britannica_1797.png height: 128 width: 360 description: no alt description found |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/1d53c38f1bbb1969092aeb2a950a315212ee7f8b height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}\log _{10}3542&=\log _{10}(1000\cdot 3.542)\\&=3+\log _{10}3.542\\&\approx 3+\log _{10}3.54\end{aligned}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/2394d5653802e38bea193035e2ec105a2399de94 height: height attribute not set width: width attribute not set description: {\displaystyle \log _{10}3542\approx {}3+\log _{10}3.54+0.2(\log _{10}3.55-\log _{10}3.54)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/961312f47e633d6119627a74dbbd40fd6b284b9a height: height attribute not set width: width attribute not set description: {\displaystyle cd=10^{\,\log _{10}c}\,10^{\,\log _{10}d}=10^{\,\log _{10}c\,+\,\log _{10}d}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/55c9e13cc3cd58305405feeae0eb07d73d26cde7 height: height attribute not set width: width attribute not set description: {\displaystyle {\frac {c}{d}}=cd^{-1}=10^{\,\log _{10}c\,-\,\log _{10}d}.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/2a448af3e9527c481ac770c0d90dba4157a8aa99 height: height attribute not set width: width attribute not set description: {\displaystyle c^{d}=\left(10^{\,\log _{10}c}\right)^{d}=10^{\,d\log _{10}c}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/2cee4c3ee52a4b250c30c38836d4d58a006ce74c height: height attribute not set width: width attribute not set description: {\displaystyle {\sqrt[{d}]{c}}=c^{\frac {1}{d}}=10^{{\frac {1}{d}}\log _{10}c}.} |
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https://upload.wikimedia.org/wikipedia/commons/thumb/8/8f/slide_rule_example2_with_labels.svg/550px-slide_rule_example2_with_labels.svg.png height: 128 width: 550 description: a slide rule: two rectangles with logarithmically ticked axes, arrangement to add the distance from 1 to 2 to the distance from 1 to 3, indicating the product 6. |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc height: height attribute not set width: width attribute not set description: {\displaystyle \mathbb {r} } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/731b0a191e1eb70161af731d0d567b236457074f height: height attribute not set width: width attribute not set description: {\displaystyle \mathbb {r} _{>0}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/711753605e98f4d42b75fe61254c3b8f311a5fd3 height: height attribute not set width: width attribute not set description: {\displaystyle b^{x}=y} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/d6e100b69edcf61555f7a41633c3c34c1a969a97 height: height attribute not set width: width attribute not set description: {\displaystyle \log _{b}\colon \mathbb {r} _{>0}\to \mathbb {r} } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/4c1de92e0c333c4644f8866e417ea56e715324ec height: height attribute not set width: width attribute not set description: {\displaystyle \log _{b}(xy)=\log _{b}x+\log _{b}y.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/2ee6f0eb6e355d16f673a0d4a21705e24a227008 height: height attribute not set width: width attribute not set description: {\displaystyle f(xy)=f(x)+f(y).} |
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https://upload.wikimedia.org/wikipedia/commons/thumb/4/49/logarithm_inversefunctiontoexp.svg/220px-logarithm_inversefunctiontoexp.svg.png height: 256 width: 220 description: the graphs of two functions. |
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https://upload.wikimedia.org/wikipedia/commons/thumb/5/57/logarithm_derivative.svg/220px-logarithm_derivative.svg.png height: 143 width: 220 description: a graph of the logarithm function and a line touching it in one point. |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/c36ba5e891172556f0477a117831310bcd38869c height: height attribute not set width: width attribute not set description: {\displaystyle {\frac {d}{dx}}\log _{b}x={\frac {1}{x\ln b}}.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/c06fb9da0538b2a1eefa892ebbfbc3fffdc98bc0 height: height attribute not set width: width attribute not set description: {\displaystyle {\frac {d}{dx}}\ln f(x)={\frac {f'(x)}{f(x)}}.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/ffcaf5c8b14b232de9ff79e9ae0960ea4966bd10 height: height attribute not set width: width attribute not set description: {\displaystyle \int \ln(x)\,dx=x\ln(x)-x+c.} |
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https://upload.wikimedia.org/wikipedia/commons/thumb/d/df/natural_logarithm_integral.svg/220px-natural_logarithm_integral.svg.png height: 110 width: 220 description: a hyperbola with part of the area underneath shaded in grey. |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/ab90a884aa3cc91d3cdcfb9b39992598131621e1 height: height attribute not set width: width attribute not set description: {\displaystyle \ln t=\int _{1}^{t}{\frac {1}{x}}\,dx.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/eed3e6eef0b8687b84a9776500c48a2793b25017 height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}\ln(tu)&=\int _{1}^{tu}{\frac {1}{x}}\,dx\\&{\stackrel {(1)}{=}}\int _{1}^{t}{\frac {1}{x}}\,dx+\int _{t}^{tu}{\frac {1}{x}}\,dx\\&{\stackrel {(2)}{=}}\ln(t)+\int _{1}^{u}{\frac {1}{w}}\,dw\\&=\ln(t)+\ln(u).\end{aligned}}} |
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https://upload.wikimedia.org/wikipedia/commons/thumb/9/9b/natural_logarithm_product_formula_proven_geometrically.svg/500px-natural_logarithm_product_formula_proven_geometrically.svg.png height: 112 width: 500 description: the hyperbola depicted twice. the area underneath is split into different parts. |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/863f225bdfee5e6256df66cc7b5c44ba2fde8e2e height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}\ln(t^{r})&=\int _{1}^{t^{r}}{\frac {1}{x}}dx\\&=\int _{1}^{t}{\frac {1}{w^{r}}}\left(rw^{r-1}\,dw\right)\\&=r\int _{1}^{t}{\frac {1}{w}}\,dw\\&=r\ln(t).\end{aligned}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/12f50a7390e77e5beed851612314d2d03991d564 height: height attribute not set width: width attribute not set description: {\displaystyle 1+{\frac {1}{2}}+{\frac {1}{3}}+\cdots +{\frac {1}{n}}=\sum _{k=1}^{n}{\frac {1}{k}},} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/d0f1edf2104b89524c509d6cb9ea1a667251d3ac height: height attribute not set width: width attribute not set description: {\displaystyle \sum _{k=1}^{n}{\frac {1}{k}}-\ln(n),} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/75b2a724c326f7f59f71baa9788cf455a0609bdc height: height attribute not set width: width attribute not set description: {\displaystyle {\sqrt {2-{\sqrt {3}}}}} |
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https://upload.wikimedia.org/wikipedia/commons/thumb/8/88/logarithm_keys.jpg/220px-logarithm_keys.jpg height: 165 width: 220 description: no alt description found |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/6d064349793d675f2707a5142e3530bd32facbdc height: height attribute not set width: width attribute not set description: {\displaystyle \log _{2}\left(x^{2}\right)=2\log _{2}|x|.} |
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https://upload.wikimedia.org/wikipedia/commons/thumb/0/02/taylor_approximation_of_natural_logarithm.gif/220px-taylor_approximation_of_natural_logarithm.gif height: 136 width: 220 description: an animation showing increasingly good approximations of the logarithm graph. |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/18c57a960210bc2a905785a5be6ba3d4ef2cfa62 height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}\ln(z)&={\frac {(z-1)^{1}}{1}}-{\frac {(z-1)^{2}}{2}}+{\frac {(z-1)^{3}}{3}}-{\frac {(z-1)^{4}}{4}}+\cdots \\&=\sum _{k=1}^{\infty }(-1)^{k+1}{\frac {(z-1)^{k}}{k}}.\end{aligned}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/6741aff58f21f04c60cd1f9e7cc2ed895d990327 height: height attribute not set width: width attribute not set description: {\displaystyle (z-1),\ \ (z-1)-{\frac {(z-1)^{2}}{2}},\ \ (z-1)-{\frac {(z-1)^{2}}{2}}+{\frac {(z-1)^{3}}{3}},\ \ldots } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/b6eb8f63f6959cb0e21a2ea79b0edd5d698f1528 height: height attribute not set width: width attribute not set description: {\displaystyle \ln(1+z)=z-{\frac {z^{2}}{2}}+{\frac {z^{3}}{3}}-\cdots \approx z.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/6e4774ed055db87556b991ffc8dcf5bd795f823c height: height attribute not set width: width attribute not set description: {\displaystyle \ln(z)=2\cdot \operatorname {artanh} \,{\frac {z-1}{z+1}}=2\left({\frac {z-1}{z+1}}+{\frac {1}{3}}{\left({\frac {z-1}{z+1}}\right)}^{3}+{\frac {1}{5}}{\left({\frac {z-1}{z+1}}\right)}^{5}+\cdots \right),} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/1d9729501b26eb85764942cb112cc9885b1a6cca height: height attribute not set width: width attribute not set description: {\displaystyle \ln(z)=2\sum _{k=0}^{\infty }{\frac {1}{2k+1}}\left({\frac {z-1}{z+1}}\right)^{2k+1}.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/70dc2d5fb51bc065e7662ba91fe25996896dfa2e height: height attribute not set width: width attribute not set description: {\displaystyle a={\frac {z}{\exp(y)}},} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/4ea935537fad8ed9632a4e0eaa453c172906606c height: height attribute not set width: width attribute not set description: {\displaystyle \ln(z)=y+\ln(a).} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/b4197236e535dc6467df2570a52e72ec095f7fd2 height: height attribute not set width: width attribute not set description: {\displaystyle \textstyle z={\frac {n+1}{n}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/306cedcb8ef32fe57d535b3c27b2ae6af9b13326 height: height attribute not set width: width attribute not set description: {\displaystyle \ln(n+1)=\ln(n)+2\sum _{k=0}^{\infty }{\frac {1}{2k+1}}\left({\frac {1}{2n+1}}\right)^{2k+1}.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/f6c094c1c61f349b1216928c33fe29aa61860626 height: height attribute not set width: width attribute not set description: {\textstyle \left({\frac {1}{2n+1}}\right)^{2}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/ec803ea11552f9cdfd17caf1b39cf8e7a8e84184 height: height attribute not set width: width attribute not set description: {\displaystyle \ln(x)\approx {\frac {\pi }{2\,\mathrm {m} \!\left(1,2^{2-m}/x\right)}}-m\ln(2).} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/966df76a5bc207e11606ad5c8ea6788d6a838c47 height: height attribute not set width: width attribute not set description: {\textstyle {\sqrt {xy}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/80443424cc40c061dba53d32f86d2d8169aa1983 height: height attribute not set width: width attribute not set description: {\displaystyle x\,2^{m}>2^{p/2}.\,} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40 height: height attribute not set width: width attribute not set description: {\displaystyle k} |
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https://upload.wikimedia.org/wikipedia/commons/thumb/0/08/nautiluscutawaylogarithmicspiral.jpg/220px-nautiluscutawaylogarithmicspiral.jpg height: 166 width: 220 description: a photograph of a nautilus' shell. |
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https://upload.wikimedia.org/wikipedia/commons/thumb/4/4f/germany_hyperinflation.svg/220px-germany_hyperinflation.svg.png height: 257 width: 220 description: a graph of the value of one mark over time. the line showing its value is increasing very quickly, even with logarithmic scale. |
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https://upload.wikimedia.org/wikipedia/commons/thumb/a/ae/pdf-log_normal_distributions.svg/220px-pdf-log_normal_distributions.svg.png height: 220 width: 220 description: three asymmetric pdf curves |
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https://upload.wikimedia.org/wikipedia/commons/thumb/1/13/benfords_law_illustrated_by_world%27s_countries_population.svg/220px-benfords_law_illustrated_by_world%27s_countries_population.svg.png height: 165 width: 220 description: a bar chart and a superimposed second chart. the two differ slightly, but both decrease in a similar fashion. |
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https://upload.wikimedia.org/wikipedia/commons/thumb/f/fb/chaotic_bunimovich_stadium.svg/220px-chaotic_bunimovich_stadium.svg.png height: 110 width: 220 description: an oval shape with the trajectories of two particles. |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/4be67693caef12b846ed9cd173a0e7a340364d27 height: height attribute not set width: width attribute not set description: {\displaystyle s=-k\sum _{i}p_{i}\ln(p_{i}).\,} |
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https://upload.wikimedia.org/wikipedia/commons/thumb/3/37/sierpinski_dimension.svg/400px-sierpinski_dimension.svg.png height: 58 width: 400 description: parts of a triangle are removed in an iterated way. |
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https://upload.wikimedia.org/wikipedia/commons/thumb/1/13/4octaves.and.frequencies.svg/350px-4octaves.and.frequencies.svg.png height: 38 width: 350 description: four different octaves shown on a linear scale. |
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https://upload.wikimedia.org/wikipedia/commons/thumb/3/31/4octaves.and.frequencies.ears.svg/350px-4octaves.and.frequencies.ears.svg.png height: 45 width: 350 description: four different octaves shown on a logarithmic scale |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/55acf246da64ba711e1717eb43ad81792220ab32 height: height attribute not set width: width attribute not set description: {\displaystyle {\frac {466}{440}}\approx {\frac {493}{466}}\approx 1.059\approx {\sqrt[{12}]{2}}.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538 height: height attribute not set width: width attribute not set description: {\displaystyle r} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/dcce5a9aa9e216fc208a597f74d2d2b6248663e6 height: height attribute not set width: width attribute not set description: {\displaystyle 2^{\frac {1}{72}}\approx 1.0097} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/18664eeb9dbe129067dd89295c4928d54fda5f01 height: height attribute not set width: width attribute not set description: {\displaystyle 2^{\frac {1}{12}}\approx 1.0595} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/84da6d3ba8b361f151624067f68d609526e6e47b height: height attribute not set width: width attribute not set description: {\displaystyle {\tfrac {5}{4}}=1.25} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/76610ca7878ea438fa73bd50ac4df1fecce09b9f height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}2^{\frac {4}{12}}&={\sqrt[{3}]{2}}\\&\approx 1.2599\end{aligned}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/fa178821ca7a1554106bf2244d08577f6f5d17fb height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}2^{\frac {6}{12}}&={\sqrt {2}}\\&\approx 1.4142\end{aligned}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/f0b7cc906bb2bc2787e32f7d1643b290d7b96766 height: height attribute not set width: width attribute not set description: {\displaystyle 2^{\frac {12}{12}}=2} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/e0bf873a44ac093c185619b4fc8e9852d569eaf3 height: height attribute not set width: width attribute not set description: {\displaystyle 12\log _{2}r} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/5bc02e655226b1a0e18922e932efff50531c48eb height: height attribute not set width: width attribute not set description: {\displaystyle {\tfrac {1}{6}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf height: height attribute not set width: width attribute not set description: {\displaystyle 1} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/c58cbdc0681a74e9ebeb1b0ce4b25833bc75cee8 height: height attribute not set width: width attribute not set description: {\displaystyle \approx 3.8631} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/295b4bf1de7cd3500e740e0f4f0635db22d87b42 height: height attribute not set width: width attribute not set description: {\displaystyle 4} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/39d81124420a058a7474dfeda48228fb6ee1e253 height: height attribute not set width: width attribute not set description: {\displaystyle 6} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/a522d3aa5812a136a69f06e1b909d809e849be39 height: height attribute not set width: width attribute not set description: {\displaystyle 12} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/dcbcf502f2dd2fba6a0f6f6758ec0707c62484ed height: height attribute not set width: width attribute not set description: {\displaystyle 1200\log _{2}r} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/a14ffb45717ec74dc340583a93d6a788f6179382 height: height attribute not set width: width attribute not set description: {\displaystyle 16{\tfrac {2}{3}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/0572cd017c6d7936a12737c9d614a2f801f94a36 height: height attribute not set width: width attribute not set description: {\displaystyle 100} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/0309fcfcbe7d98188d148b01f6f41d15ea416d2c height: height attribute not set width: width attribute not set description: {\displaystyle \approx 386.31} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/8540670f7baa60a08a5dd4b12916c16fe6faf200 height: height attribute not set width: width attribute not set description: {\displaystyle 400} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/5ed5fbc94ba594303754ec8efd3d552547a93043 height: height attribute not set width: width attribute not set description: {\displaystyle 600} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/973054497debca94837d3a844349fe9221727dbd height: height attribute not set width: width attribute not set description: {\displaystyle 1200} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/3e7c35556de976b4896475a163d924bb9f3d83eb height: height attribute not set width: width attribute not set description: {\displaystyle {\frac {x}{\ln(x)}},} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/da577950b8b4c4ef726a3065afcdafa378dbc3fb height: height attribute not set width: width attribute not set description: {\displaystyle \mathrm {li} (x)=\int _{2}^{x}{\frac {1}{\ln(t)}}\,dt.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/207af1498eca9a74c5e19ecab897d915d1052c95 height: height attribute not set width: width attribute not set description: {\displaystyle \ln(n!)=\ln(1)+\ln(2)+\cdots +\ln(n).} |
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https://upload.wikimedia.org/wikipedia/commons/thumb/8/83/complex_number_illustration_multiple_arguments.svg/220px-complex_number_illustration_multiple_arguments.svg.png height: 234 width: 220 description: an illustration of the polar form: a point is described by an arrow or equivalently by its length and angle to the x-axis. |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/7dc3322ef38b06276c3bee59d656769b6edf531f height: height attribute not set width: width attribute not set description: {\displaystyle e^{a}=z} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/0c0606ea41983c11ed8e73fc1507ce58ad316ef0 height: height attribute not set width: width attribute not set description: {\displaystyle \textstyle r={\sqrt {x^{2}+y^{2}}}.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/b6851b66ce18570803cf1ffdc5b789016559b218 height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}z&=x+iy\\&=r(\cos \varphi +i\sin \varphi )\\&=r(\cos(\varphi +2k\pi )+i\sin(\varphi +2k\pi )),\end{aligned}}} |
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https://upload.wikimedia.org/wikipedia/commons/thumb/a/af/complex_log_domain.svg/220px-complex_log_domain.svg.png height: 165 width: 220 description: a density plot. in the middle there is a black point, at the negative axis the hue jumps sharply and evolves smoothly otherwise. |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/396158ab1664889849843ce26a324ed8dbbf841e height: height attribute not set width: width attribute not set description: {\displaystyle e^{i\varphi }=\cos \varphi +i\sin \varphi .} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/0a7f91b7eb61729d49185f2c48d2eb32c40a11a3 height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}z&=r\left(\cos \varphi +i\sin \varphi \right)\\&=r\left(\cos(\varphi +2k\pi )+i\sin(\varphi +2k\pi )\right)\\&=re^{i(\varphi +2k\pi )}\\&=e^{\ln(r)}e^{i(\varphi +2k\pi )}\\&=e^{\ln(r)+i(\varphi +2k\pi )}=e^{a_{k}},\end{aligned}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/22659f325d07ee703329595fc7cf9d372c4fb086 height: height attribute not set width: width attribute not set description: {\displaystyle a_{k}=\ln(r)+i(\varphi +2k\pi ),} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/f90e5e0851759d0490e085cf1888338465384143 height: height attribute not set width: width attribute not set description: {\displaystyle b^{n}=x,} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/c16a216f9168ba23df2d07ceb32c6929a70c4e1b height: height attribute not set width: width attribute not set description: {\displaystyle \operatorname {li} _{s}(z)=\sum _{k=1}^{\infty }{z^{k} \over k^{s}}.} |
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https://upload.wikimedia.org/wikipedia/commons/thumb/3/3e/nuvola_apps_edu_mathematics_blue-p.svg/28px-nuvola_apps_edu_mathematics_blue-p.svg.png height: 28 width: 28 description: icon |
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https://upload.wikimedia.org/wikipedia/commons/thumb/a/a3/arithmetic_symbols.svg/28px-arithmetic_symbols.svg.png height: 28 width: 28 description: no alt description found |
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https://upload.wikimedia.org/wikipedia/commons/thumb/e/ed/papapishu-lab-icon-6.svg/28px-papapishu-lab-icon-6.svg.png height: 28 width: 28 description: no alt description found |
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https://upload.wikimedia.org/wikipedia/commons/thumb/f/f5/terra.png/28px-terra.png height: 28 width: 28 description: icon |
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https://upload.wikimedia.org/wikipedia/commons/thumb/7/7a/nuvola_apps_kcmsystem.svg/28px-nuvola_apps_kcmsystem.svg.png height: 28 width: 28 description: icon |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/361b01fa020a096dc96de0f7b07de6eadba74597 height: height attribute not set width: width attribute not set description: {\textstyle x=b^{\log _{b}x},} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/f9963d4106a92dfa004d0110b8415407fa52ce9c height: height attribute not set width: width attribute not set description: {\displaystyle \log _{k}x=\log _{k}\left(b^{\log _{b}x}\right)=\log _{b}x\cdot \log _{k}b.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/b8f58e6c48d751c89792f8c702a8a2537f1621d9 height: height attribute not set width: width attribute not set description: {\displaystyle \log _{b}x.} |
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https://upload.wikimedia.org/wikipedia/en/thumb/4/4a/commons-logo.svg/12px-commons-logo.svg.png height: 16 width: 12 description: no alt description found |
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https://upload.wikimedia.org/wikipedia/commons/thumb/9/99/wiktionary-logo-en-v2.svg/16px-wiktionary-logo-en-v2.svg.png height: 16 width: 16 description: no alt description found |
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https://upload.wikimedia.org/wikipedia/commons/thumb/f/fa/wikiquote-logo.svg/13px-wikiquote-logo.svg.png height: 16 width: 13 description: no alt description found |
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https://upload.wikimedia.org/wikipedia/commons/thumb/0/0b/wikiversity_logo_2017.svg/16px-wikiversity_logo_2017.svg.png height: 13 width: 16 description: no alt description found |
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https://upload.wikimedia.org/wikipedia/en/thumb/8/8a/oojs_ui_icon_edit-ltr-progressive.svg/10px-oojs_ui_icon_edit-ltr-progressive.svg.png height: 10 width: 10 description: edit this at wikidata |
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https://login.wikimedia.org/wiki/special:centralautologin/start?type=1x1 height: 1 width: 1 description: no alt description found |
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http://en.wikipedia.org/static/images/footer/wikimedia-button.svg height: 29 width: 84 description: wikimedia foundation |
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http://en.wikipedia.org/w/resources/assets/poweredby_mediawiki.svg height: 31 width: 88 description: powered by mediawiki |
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