the number e is ________________ it is also known as euler’s number irrational
TRANSCRIPT
xxe lnlog
...!4
1
!3
1
!2
1
!1
1
!0
1e
...24
1
6
1
2
1
1
1
1
1e
7182818284.2eThe number e is ________________
It is also known as Euler’s number
Irrational
Domain:Range:Y-intercept:Asymptote:
X Y
-1
0
1
2
Type of Function::
4.718.2
11
e
10 e
718.21 ee
Growth
All reals xY>0
(0,1)Y=0
4.72 e
Domain:Range:Y-intercept:Asymptote:
X Y
-1012
Type of Function::
4.718.2
11)1(
ee
100 ee7.2)1( ee
Decay All reals xY>0(0,1)Y=0
f(x) =eax, rep exp growth when a is ___________ and decay when a is _________
Positivenegative
4.72 e
SOLVE for x in the following
xee 5
x5
xeee 32 xee 24
xee 8
x8Use the calculator and
round 3 places
_____3 ee ________33 ee 086.20
Change the following exponential fcns to logarithmic form
1. 2. 8134 xe 2
481log3 2log xe
In the next slide
you will see
why this can
also be written
as a natural log
If x is a positive real number, then the natural logarithm of x is denoted by:
xelog or xln
Note that if a base is not
written here – it is base e
Calculators can evaluate logs with the
common base….. which is base 10
They can also evaluate the natural
logarithm…which is base e , the natural number
Use the log key
Use the ln key
We will do more
calculator log problems in 8.5
These 2 graphs are reflections over the line _________ y = x
Exp fcn
HA: y=o VA: x=o
Natural log fcn
Is ________ ofinverse
Special Values of Logarithms
ln 1 =___
?=0
1? e
ln e =___ ln ex =___ eln(x) =___
?1log e
0?log ee
ee ?
?=1
?ln eX
?1xX=?
1 X
?=x
)ln((?)log xe
)(log(?)log xee
Hint: HW 12-13
x
ea·ln(x) =___
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SOLVE for x using the properties of logs
ln 12 = ln 3 + ln x
)3ln(12ln xx312x4
SOLVE for x using the properties of logs
5 ln(x) = 20
4ln x4ln ee x
598.544 xorex
SOLVE for x using the properties of logs
5ex=10
ex=2
ln ex = ln 2
x = ln 2 or x = 0.693
Log fcn has VA: no horizontal shift so x=o
Can also be written as:
This is a log fcn with base e and is to be shifted:
The negative causes a reflection in the x axis. The 2 cause a vertical shift up 2
Remember: ln 1 = 0
X Y
1 2
Activity: The following are actually for lesson 8.5
HW : WS 8.4 – which is due next class
___ln e
xee log
eex
___1ln
xe 1log
1xe
x37 21
x37
x3
7
X=0.882
126 xe2xexe 2log
X=0.693xe
log
2logUse the change of base formula
Or more commonly seen
Take ln of both sidesTo eliminate the base e
2lnln xe
5
14 xe
20
1xe
20
1lnln xe
X=-2.996Take ln of both sidesTo eliminate the base e
622
1x
122 x
x12log2
x2log
12logX=3.585
Use the change of base formula
Convert to a log
Does this equalLog 6
245 2 x
224log5 x
x 25log
24log X=3.975
19ln6 x
6
19log xe
xe 6/19X=23.728
6
19ln x
Convert to a log
Convert to an exponential
Or more commonly seen
6
19ln ee x
10ln5 x2ln x2log xexe 2 X=7.389
or
2ln ee x
27log3
24log
2
1log mmm x
X=18
9log2loglog mmm x
18loglog mm x