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1.5 Continued... Logarithms

Base a Logarithm Fu= nction

Base a Logarithm Fu= nction

log(x) and ln(x)

Properties of logar= ithms

Example 4 (Similar)=

Extra Example

More Properties of Logarithms

Changing Bases<= /b>

Example 5 (Similar)=

Applications

Applications

Example 7

Assignment<= /font>

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1.5 Continued... Logarithms

Base a Logarithm Fu= nction
If a is any positiv= e real number other than one, the base a exponential function f(x)=3Dax is one-to-one.  Therefore, it has an inverse.
This inverse is wha= t we call the base a logarithm function.

Base a Logarithm Fu= nction
Definition: The bas= e a logarithm function y =3D logax is the inverse of the base a exponential function y=3Dax (a > 0, a≠1)
Domain: The domain = of y =3D logax is (0, ∞), the range of ax.
Range: The range of= y =3D logax  is   (-∞,∞), the domain of ax.

log(x) and ln(x)
Logarithms with bas= e 10 and base e are used most commonly, so they have their own special notation and names.
logex = =3D lnx à natural lo= g
log10x = =3D logx à common log=

Properties of logar= ithms
Inverse Properties:=
Base a:
alogax =3D x, a > 1, x > 0
loga ax =3D x, x>0
Base e:
eln x = =3D x, x > 0
ln ex = =3D x

Example 4 (Similar)=
Solve for x.=
a) ln x =3D 8y= 2 - 3
b) e2x = =3D 8

Extra Example
Solve for x: 4= x + 4-x =3D 10

More Properties of Logarithms
For any real number= s x>0 and y>0,
Product Rule:
loga xy =3D loga x + loga y
Quotient Rule:
loga (x/= y) =3D loga x – loga y
Power Rule:<= /td>
loga xy =3D y loga x

Changing Bases<= /b>
If we need to evalu= ate loga x for any base a>0, a≠1, we can use the following formula:=
Change of Base Form= ula:

Example 5 (Similar)=
Graph f(x) =3D log<= sub>7 x
Hint: Use the formu= la                        and your graphi= ng calculator.

Applications
You invest $2000 in= an account that earns 4.75% interest compounded annually.  How long will it take until your mone= y is doubled?

Applications
The population of S= eneca is 2100 and increasing by 1.65% each year. Predict when the population will reach 10,000.

Example 7
The table shows the= number of metric tons of oil produced by Indonesia for 3 different years.
Find the natural lo= garithm regression equation for the data in the table and use it to estimate the number of metric tons of oil produced by Indonesia in 1982 and 2000.
Enter data in L1 an= d L2
STAT,>Calc, 9 (L= nReg)
Y1=3DVARS, 5, >&= gt;EQ, 1 RegEQ
Set up window &= graph
2nd CALC= , 1 to check for 1982 and 2000

Assignment<= /font>
Sec 1.5: 37-49
Follow directions
Solve algebraically= unless directed otherwise

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=
1.5 Continued...
L= ogarithms
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= Base a Logarithm Function
<= /td> <= /td> = = <= /td> <= /td>
n If = a is any positive real number other than
one= , the base a exponential function
= f(x)=3Dax is one-to-one.  Therefore, it h= as an
inve= rse.
n Thi= s inverse is what we call the base a
= logarithm function.
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= Base a Logarithm Function
<= /td> = <= /td> = <= /td> <= /td>
n = Definition: The base a logarithm function
= y =3D logax is the inverse of the base a
exp= onential function y=3Dax (a > 0, a≠1)
n Domain: The domain of y =3D logax is (0, ∞= ;),
the= range of ax.
n Range: The range of y =3D logax  is=    (-∞,∞),
the= domain of ax.
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= log(x) and ln(x)
<= /td> = <= /td> <= /td> <= /td>
n Log= arithms with base 10 and base e are
use= d most commonly, so they have their
own= special notation and names.
n logex =3D lnx natural log
n log10x =3D logx common log
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= Properties of logarithms
<= /td> <= /td> <= /td> <= /td> <= /td> <= /td> <= /td>
n Inv= erse Properties:
n Base= a:
¨ alogax =3D x, a > 1, x > 0
¨ loga ax =3D x, x>0
n Base= e:
¨ eln x =3D x, x > 0
¨ ln ex =3D x
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= Example 4 (Similar)
<= /td> <= /td> <= /td>
n Sol= ve for x.
a) = ln x =3D 8y2 - 3
b) <= i>e2x =3D 8
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= Extra Example
<= /td>
n Solve for x: 4x + 4-x =3D 10
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= More Properties of Logarithms
<= /td> <= /td> = <= /td> <= /td> <= /td> <= /td>
n Fo= r any real numbers x>0 and y>0,
n Pro= duct Rule:
¨ loga xy =3D loga x + loga y
n Quo= tient Rule:
¨ loga (x/y) =3D loga x – log= a y
n Pow= er Rule:
¨ loga xy =3D y loga <= /i>x
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= Changing Bases
<= /td> = =
n If we need to evaluate loga x for any base a&= gt;0,
= a≠1, we can use the following formula:
n Cha= nge of Base Formula:
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Example 5   (Similar)
<= /td> = <= /td>
n Graph f(x) =3D log7 x
n Hin= t: Use the formula                        and
you= r graphing calculator.
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file:///C:/C95ACE62/CalcCh1Sec5Part21B_files/v3_slide0011.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="windows-1252" 1.5 B Logarithms
Applications
<= /td> = =
n You= invest $2000 in an account that earns 4.75%
int= erest compounded annually.  How long = will it take until
you= r money is doubled?
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Applications
<= /td> <= /td> =
n The= population of Seneca is 2100 and increasing by
1.6= 5% each year. Predict when the population will reach
10,0= 00.
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Example 7
<= /td> <= /td> <= /td> <= /td> <= /td> <= /td> <= /td> <= /td>
n Th= e table shows the number of
met= ric tons of oil produced by
In= donesia for 3 different years.
n Fin= d the natural logarithm
re= gression equation for the data in
th= e table and use it to estimate the
num= ber of metric tons of oil
pr= oduced by Indonesia in 1982
and = 2000.
n Ent= er data in L1 and L2
n STA= T,>Calc, 9 (LnReg)
n Y1= =3DVARS, 5, >>EQ, 1 RegEQ
n Set= up window & graph
n 2nd CALC, 1 to check for 1982 and
2000=
<= /td> <= /td>
Indonesia’s Oil
P= roduction
<= /td>
M= etric
Tons-in
millions
Y= ear
1= 960
2= 0.56
1= 970
4= 2.10
1= 990
7= 0.10
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= Assignment
<= /td> <= /td> = <= /td>
n Sec= 1.5: 37-49
¨ Fol= low directions
¨ Sol= ve algebraically unless directed otherwise
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1.5 Continued... = Logarithms
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file:///C:/C95ACE62/CalcCh1Sec5Part21B_files/slide0002.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" 1.5 B Logarithms
Base a Logarithm Function
nIf a is any positive real nu= mber other than one, the base a exponential func= tion f(x)=3Dax is one-to-one.  Therefore, it has an inverse.
nThis inverse is what we call the base a logarithm function.
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Base a Logarithm Function
nDefinition: The base a logarithm function y =3D logax is the inverse of the base a exponential function = y=3Dax (a > 0, a≠1) <= /div>
<= span class=3DBB style=3D'position:absolute;left:-3.78%;top:.3em'>nDomain= : The domain of y =3D logax is (0, ∞), the range of ax.
<= span class=3DBB style=3D'position:absolute;left:-3.81%;top:.3em'>nRange:= The range of y =3D logax  is   (-∞,∞), = the domain of ax.
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log(x) and ln(x)
nLogarithms with base 10 and base e are used most commonly, so they have their own special notation and names.
<= span class=3DBB style=3D'position:absolute;left:-4.43%;top:.3em'>nlog= ex =3D lnx = à natural log
<= span class=3DBB style=3D'position:absolute;left:-4.43%;top:.3em'>nlog= 10x =3D logx &a= grave; common log
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Properties of logarithms
<= span class=3DBB style=3D'position:absolute;left:-4.39%;top:.3em'>nInverse Properties:
<= span class=3DBB style=3D'position:absolute;left:-4.39%;top:.3em'>nBase a= :
<= span class=3DB1B style=3D'position:absolute;left:-3.93%;top:.22em'>¨= alogax =3D x, a > 1, x > 0 =
<= span class=3DB1B style=3D'position:absolute;left:-3.93%;top:.22em'>¨= loga ax =3D x, x>0
<= span class=3DBB style=3D'position:absolute;left:-4.39%;top:.3em'>nBase e= :
<= span class=3DB1B style=3D'position:absolute;left:-3.93%;top:.22em'>¨= eln= x =3D x, x > 0
<= span class=3DB1B style=3D'position:absolute;left:-3.93%;top:.22em'>¨= ln ex =3D x
¨
¨
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Example 4 (Similar)
<= span class=3DBB style=3D'position:absolute;left:-4.43%;top:.3em'>nSolve = for x.
na) ln x =3D 8y2<= /span> - 3
n
n
nb) e2x =3D 8
n
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Extra Example
nSolve for x: 4x + 4-x =3D 10
<= img border=3D0 v:shapes=3D"SMARTPenAnnotation5842,SMARTPenAnnotation5843,SMARTPenAnnotati= on5844,SMARTPenAnnotation5845,SMARTPenAnnotation5846,SMARTPenAnnotation5847= ,SMARTPenAnnotation5848,SMARTPenAnnotation5849,SMARTPenAnnotation5850,SMART= PenAnnotation5851,SMARTPenAnnotation5852,SMARTPenAnnotation5853,SMARTPenAnn= otation5854" src=3D"slide0007_image037.gif" style=3D'position:absolute;top:26.5%;left:6= 9.85%; width:27.71%;height:7.0%'>
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