the logarithmic function with base
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THE LOGARITHMIC FUNCTION WITH BASE b is thefunction
y = logb x
b is no!"#ll$ # nu"%e! g!e#te! th#n & '#lthough it nee(onl$ %e g!e#te! th#n ) #n( not e*u#l to &+ The function is(e,ne( fo! #ll x - ) He!e is its g!#.h fo! #n$ %#se b
Note the follo/ing0
1 Fo! #n$ %#se2 the x3inte!ce.t is & Wh$4To see the answer, pass your mouse over thecolored area.To cover the answer again, click "Refresh"("Reload").
The logarithm of 1 is 0. y = logb1= 0.
1 The g!#.h .#sses th!ough the .oint 'b2 &+ Wh$4
The logarithm of the base is 1. logbb = 1.
1The g!#.h is %elo/ the x3#5is 33 the log#!ith" isneg#ti6e 33 fo!
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) 7 x 7 &
Which nu"%e!s #!e those th#t h#6e neg#ti6elog#!ith"s4
Proper fractions.
1The function is (e,ne( onl$ fo! .ositi6e 6#lues of x
logb'89+2 fo! e5#".le2 "#:es no sense Since b is #l/#$s.ositi6e2 no .o/e! of b c#n .!o(uce # neg#ti6e nu"%e!
1 The !#nge of the function is #ll !e#l nu"%e!s
1 The neg#ti6e y 3#5is is # 6e!tic#l #s$".tote 'To.ic&;+
Example 1. Translation of axes. He!e is the g!#.h ofthe n#tu!#l log#!ith"2 y = ln x 'To.ic <)+
An( he!e is the g!#.h of y = ln 'x 8 <+ 33 /hich isits translation < units to the !ight
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The x3inte!ce.t h#s "o6e( f!o" & to An( the 6e!tic#l #s$".tote h#s "o6e( f!o" ) to <
Problem 1. S:etch the g!#.h of y = ln 'x > +
This is a translation 3 units to the left. The x -intercept has moved from 1 to −. !nd the verticalas"mptote has moved from 0 to −3.
Exponential functions
B$ (e,nition0
logb y = x "e#ns bx = y
In othe! /o!(s2 co!!es.on(ing to e6e!$ log#!ith"function /ith %#se bthe!e is #n e5.onenti#l function /ith%#se b0
y = bx
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It is (e,ne( fo! e6e!$ !e#l nu"%e! x He!e is itsg!#.h0
The!e #!e t/o i".o!t#nt things to note01 The y 3inte!ce.t is #t ')2 &+ Fo!2 b) = &
1 The neg#ti6e x3#5is is # ho!i?ont#l #s$".tote Fo!2/hen x is # l#!ge neg#ti6e nu"%e! 33 eg b8&)2))) 33 then y is # 6e!$ s"#ll .ositi6e nu"%e!
Problem .
#+ Let f 'x+ = ex W!ite the function f '8x+
f #−x $ = e− x
The argument x is replaced b" − x .
%+ Wh#t is the !el#tionshi. %et/een the g!#.h of y = ex #n( the g!#.h%+ of y = e8x 4
y = e−x
is the re%ection about the y-axis of y = e x .
c+ S:etch the g!#.h of y = e8x
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Inverse relations
E5.onenti#l functions #n( log#!ith"ic functions /ith%#se b #!e in6e!ses
The functions logbx #n( bx #!e in6e!ses
Fo! in #n$ %#se b0
i+ blogbx = x2
#n(
ii+ logbbx = x
Rule i+ e"%o(ies the (e,nition of # log#!ith"0 logbx isthe exponentto /hich b "ust %e !#ise( to .!o(uce x
Rule ii+ /e h#6e seen in the .!e6ious To.ic
No/2 let
f 'x+ = bx #n( g'x+ = logbx
Then Rule i+ is f 'g'x++ = x
An( Rule ii + is g' f 'x++ = x
These !ules s#tisf$ the (e,nition of # .#i!of in6e!se functions 'To.ic &@+ The!efo!e fo! #n$ %#se b2the functions
f 'x+ = bx #n( g'x+ = logbx
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#!e in6e!ses
Problem 3. E6#lu#te the follo/ing
#+
log<< = &
%+ log
&).<
= '.
c+ ln ex >
&
= x (
1
(+ <log<
= & e+ &)log &)) = 100
f+ eln 'x 8
+= x −&
Problem ).
#+ Wh#t function is the in6e!se of y = ln x4
y =
e
x
.%+ Let f 'x+ = ln x #n( g'x+ = ex2 #n( sho/th#t f #n( g s#tisf$ the%+ in6e!se !el#tions
f #g# x $$ = lne x = x *
g# f # x $$ = eln
x
= x .
He!e #!e the g!#.hs of y = ex #n( y = ln x 0
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As /ith #ll .#i!s of in6e!se functions2 thei! g!#.hs#!e s$""et!ic#l /ith !es.ect to the line y = x 'See To.ic&@+
Problem &. E6#lu#te ln e#!ccos '8&+
ln earccos #−1$ = arccos #−1$ = +.
,The angle hose cosine is −1
is +.,
See To.ic &@ of T!igono"et!$
Exponential and logarithmic equations
Example . Sol6e this e*u#tion fo! x 0
x > & = <
olution To !ele#se x > & f!o" the e5.onent2 t#:e thein6e!se function 33 the log#!ith" /ith %#se 33 of %othsi(es E*ui6#lentl$2 /!ite the log#!ith"ic fo!" 'To.ic <)+
log-x > & = log<
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x > & = log<
x > & = 9
x =
Example 3. Sol6e fo! x 0
<x 8 9 = x
olution We "#$ t#:e the log of %oth si(es eithe! /iththe %#se < o! the %#se Let us use %#se <0
log<<x 8 9 =
log<x
x 8 9 =
x log<2 #cco!(ing to the !(L#/
x 8 x log< =
9
x'& 8 log<+ =
9
x =
9& 8 log<
log< is so"e nu"%e! The e*u#tion is sol6e(
Problem '. Sol6e fo! x 0
<x 8 = <
log x − & = log3
x − & = &
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x = 10
Problem /. Sol6e fo! x The solution "#$ %e e5.!esse(#s # log#!ith"
&)x 8 & = <<x > &
log 103 x − 1 = log x ( 1
3 x − 1 = # x ( 1$ log
3 x − 1 = x log ( log
3 x − x log = 1 ( log
x #3 − log $ = 1 ( log
x =1 ( log 3 − log
Problem . Sol6e fo! x 0
esin x = &
ln esin x = ln 1
sin x = 0
x is the radian angle hose sine is 0
x = 0.
Example ). Sol6e fo! x0
log'<x > + = olution. To f!ee the #!gu"ent of the log#!ith"2 t#:ethe in6e!se function 33 x 33 of %oth si(es Th#t is2 let e#chsi(e %e the e5.onent /ith %#se E*ui6#lentl$2 /!itethe e5.onenti#l fo!"
<x > =
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<x = &< 8
<x = &<<
x = &
Problem 2. Sol6e fo! x 0
log9'x 8 + = )
f e let each side be the exponent ith base )*then
3 x − & =
)0 = 1
3 x =
'
x =
Problem 10. Sol6e fo! x 0
log<'xD > + = 9
x 4 ( / = ) = 1'
x 4 = 1' − / = 2
x = 53
Example &. Sol6e fo! x0
log '<x > &+ = log &&olution If /e let e#ch si(e %e the e5.onent /ith &) #sthe %#se2 then #cco!(ing to the in6e!se !el#tions0
<x > &= &&
Th#t i".lies
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x=
Problem 11. Sol6e fo! x0
ln 'x 8 &+ = ln '<x > ;+
f e let each side be the exponent ith base e*then
& x − 1 = x (
3 x = 2
x = 3.
S:ill in Alge%!#2 Lesson @
Creating one logarithm from a sum
Example '. Use the l#/s of log#!ith"s 'To.ic <)+ to/!ite the follo/ing #s one log#!ith"
log x > log y 8 < log
olution log x > log y 8 < log = log xy 8 log D
= logxy D
Problem 1. W!ite #s one log#!ith"0
k log x > m log y 8 n log
Problem 13. W!ite #s one log#!ith"0
log '<x 8 ;+ 8 log 'xD 8 &+
log '<x 8 ;+ 8 log 'xD 8 &+ = log x − x 4 − 1'
= log # x − )$
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# x − )$# x ( )$
= log
x ( )
Example /. B$ "e#ns of Rule i #%o6e 33
n = logbbn2
33 /e c#n /!ite #n$ nu"%e! #s # log#!ith" in #n$ %#se
Fo! e5#".le2
= log<<
@ = log.@
t = ln et
= log &)))
Problem 1).
#+ < = ln e4 %+ & = ln e
Example . W!ite the follo/ing #s one log#!ith"0
logbx > n
olution. logbx > n = logbx > logbbn
= logbxbn
Problem 1&. W!ite #s one log#!ith"0
log < >
log ( 3 = log ( log 103
= log 6 103
= log 000
Problem 1'. W!ite #s one log#!ith"0
ln A 8 t
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ln ! − t = ln ! − ln et
= ln ! ( ln e−t
= ln ! e−t
Problem 1/. Sol6e fo! x0
log<x > log<'x > <+ =
log7 x # x ( $8 =
3.
f e no let each side be the exponent ith base *then
x # x ( $ =
3 = .
x 4 ( x − =
0
# x − $# x ( )$ =
0
x =
or −).
See S:ill in Alge%!#2 Lesson
We "ust !eect the solution x = 8 92 ho/e6e!2 %ec#usethe neg#ti6e nu"%e! 89 is not in the (o"#in of log<x
Problem 1. Sol6e fo! x
ln '& > x+ 8 ln '& 8 x+ =
&
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= 1.
f e no let each side be the exponent ith base e*then
= e
1 ( x = e − e x
e x ( x
=
e − 1
#e ( 1$ x = e − 1
x =