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LOGARITHIMS
Section 6.4
Sinceexponentialfunctionsareone‐to‐one,
eachhasaninverse.Theseinverse
exponentialfunctionsarecalledlogarithms.
Logarithmic Functions
LOGARITHMIC FUNCTIONS
Thelogarithmicfunctiontothebasea,
where
0 and
1,isdenotedby
log
(readas“ isthelogarithmtothebase
of ”)andisdefinedby
log
EXPONENTIAL AND
LOGARITHMIC FORMS
• Theexponentialformof
.
• Thelogarithmicformof
log .
log
is
is
ifandonlyif
Thedomainofthelogarithmicfunction
log is
0.
GRAPHING LOGARITHMIC
FUNCTIONS
Toquicklygraphthelogarithmicfunction
log
plotpointsfor
,1,anda.
1
1
a
−1
0
1
PROPERTIES OF f (x) = loga x
• Domain: 0, ∞ ;Range: ∞, ∞
• The ‐interceptofthegraphis1.Thereisno ‐
intercept.
• VerticalAsymptote:
0
• Increasingif
1
• Decreasingif0
1
• Thegraphof containsthepoints
, 1 , 1,0 ,
and , 1 .
• Thegraphissmoothandcontinuous,withno
cornersorgaps.
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DOMAIN OF A LOGARITHMIC
FUNCTIONS
Sincethelogarithmofanegativenumberand
thelogarithmofzerocannotbetaken,the
argumentofalogarithmicfunctionmust
alwaysbepositive.Thatis,ifZ isanalgebraic
expressionin ,thedomainof
log
isthesetofnumberssuchthat
0.
COMMON AND NATURAL
LOGARITHMS
Logarithmswithabaseof10arecalled
commonlogarithms.Wedenotethisby
log .Thatis,
log
log
Logarithmswithabaseofe arecalled
naturallogarithms.Wedenotethisby
ln .Thatis,
ln
log
LOGARITHMIC EQUATIONS
Equationsthatcontainlogarithmsarecalled
logarithmicequation.Somelogarithmic
equationscanbesolvedbyconvertingthemto
exponentialform.However,whensolving
logarithmicequations,youmustalwayscheck
yoursolutions.
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