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More on Log and Exponential Equations
------=_NextPart_01C957DB.BC238AB0 Content-Location: file:///C:/09764A33/5MoreonLogandExponentialEquations_files/slide0076.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" More on Log and Exponential Equations
In this s= ection we continue with log-equations and
exp-equat= ions.
More on Log and Exponential Equations
------=_NextPart_01C957DB.BC238AB0 Content-Location: file:///C:/09764A33/5MoreonLogandExponentialEquations_files/slide0077.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" More on Log and Exponential Equations
In this s= ection we continue with log-equations and
exp-equat= ions. They may be grouped in two types.
More on Log and Exponential Equations
------=_NextPart_01C957DB.BC238AB0 Content-Location: file:///C:/09764A33/5MoreonLogandExponentialEquations_files/slide0078.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" More on Log and Exponential Equations
In this s= ection we continue with log-equations and
exp-equat= ions. They may be grouped in two types. 
* Equatio= ns that do not need calculators
  (Equations with related bases on = both sides).
More on Log and Exponential Equations
------=_NextPart_01C957DB.BC238AB0 Content-Location: file:///C:/09764A33/5MoreonLogandExponentialEquations_files/slide0079.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" More on Log and Exponential Equations
In this s= ection we continue with log-equations and
exp-equat= ions. They may be grouped in two types. 
* Equatio= ns that do not need calculators
  (Equations with related bases on = both sides).
* Numeric= al equations that require calculators.
More on Log and Exponential Equations
------=_NextPart_01C957DB.BC238AB0 Content-Location: file:///C:/09764A33/5MoreonLogandExponentialEquations_files/slide0085.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" More on Log and Exponential Equations
In this s= ection we continue with log-equations and
exp-equat= ions. They may be grouped in two types. 
* Equatio= ns that do not need calculators
  (Equations with related bases on = both sides).
* Numeric= al equations that require calculators.
More on Log and Exponential Equations
Equations That Do Not Need Calculators
------=_NextPart_01C957DB.BC238AB0 Content-Location: file:///C:/09764A33/5MoreonLogandExponentialEquations_files/slide0141.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" More on Log and Exponential Equations
In this s= ection we continue with log-equations and
exp-equat= ions. They may be grouped in two types. 
* Equatio= ns that do not need calculators
  (Equations with related bases on = both sides).
* Numeric= al equations that require calculators.
More on Log and Exponential Equations
Equations That Do Not Need Calculators
The Law of Uniqueness of Log and Exp Functions= :
------=_NextPart_01C957DB.BC238AB0 Content-Location: file:///C:/09764A33/5MoreonLogandExponentialEquations_files/slide0142.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" More on Log and Exponential Equations
In this s= ection we continue with log-equations and
exp-equat= ions. They may be grouped in two types. 
* Equatio= ns that do not need calculators
  (Equations with related bases on = both sides).
* Numeric= al equations that require calculators.
More on Log and Exponential Equations
Equations That Do Not Need Calculators
If logb(U) =3D logb(V) then = U =3D V.
The Law of Uniqueness of Log and Exp Functions= :
------=_NextPart_01C957DB.BC238AB0 Content-Location: file:///C:/09764A33/5MoreonLogandExponentialEquations_files/slide0143.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" More on Log and Exponential Equations
In this s= ection we continue with log-equations and
exp-equat= ions. They may be grouped in two types. 
* Equatio= ns that do not need calculators
  (Equations with related bases on = both sides).
* Numeric= al equations that require calculators.
More on Log and Exponential Equations
Equations That Do Not Need Calculators
If logb(U) =3D logb(V) then U =3D V.
If expb(U) =3D expb(V) then U =3D V.
The Law of Uniqueness of Log and Exp Functions= :
------=_NextPart_01C957DB.BC238AB0 Content-Location: file:///C:/09764A33/5MoreonLogandExponentialEquations_files/slide0144.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" More on Log and Exponential Equations
In this s= ection we continue with log-equations and
exp-equat= ions. They may be grouped in two types. 
* Equatio= ns that do not need calculators
  (Equations with related bases on = both sides).
* Numeric= al equations that require calculators.
More on Log and Exponential Equations
Equations That Do Not Need Calculators
Use this law to simpli= fy log or exp equations when both sides can be consolidated into a common b= ase.
If logb(U) =3D logb(V) then U =3D V.
If expb(U) =3D expb(V) then U =3D V.
The Law of Uniqueness of Log and Exp Functions= :
------=_NextPart_01C957DB.BC238AB0 Content-Location: file:///C:/09764A33/5MoreonLogandExponentialEquations_files/slide0145.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" More on Log and Exponential Equations
In this s= ection we continue with log-equations and
exp-equat= ions. They may be grouped in two types. 
* Equatio= ns that do not need calculators
  (Equations with related bases on = both sides).
* Numeric= al equations that require calculators.
More on Log and Exponential Equations
Equations That Do Not Need Calculators
Use this law to simpli= fy log or exp equations when both sides can be consolidated into a common b= ase.
An exampl= e of numbers with common base are
{.., 1/8, ¼, ½, 1, 2, 4, 8,..}.
If logb(U) =3D logb(V) then U =3D V.
If expb(U) =3D expb(V) then U =3D V.
The Law of Uniqueness of Log and Exp Functions= :
------=_NextPart_01C957DB.BC238AB0 Content-Location: file:///C:/09764A33/5MoreonLogandExponentialEquations_files/slide0146.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" More on Log and Exponential Equations
In this s= ection we continue with log-equations and
exp-equat= ions. They may be grouped in two types. 
* Equatio= ns that do not need calculators
  (Equations with related bases on = both sides).
* Numeric= al equations that require calculators.
More on Log and Exponential Equations
Equations That Do Not Need Calculators
Use this law to simpli= fy log or exp equations when both sides can be consolidated into a common b= ase.
An exampl= e of numbers with common base are
{.., 1/8, ¼, ½, 1, 2, 4, 8,..}. All these numbers are = base 2 nu= mbers, i.e. they are powers of 2.
If logb(U) =3D logb(V) then U =3D V.
If expb(U) =3D expb(V) then U =3D V.
The Law of Uniqueness of Log and Exp Functions= :
------=_NextPart_01C957DB.BC238AB0 Content-Location: file:///C:/09764A33/5MoreonLogandExponentialEquations_files/slide0053.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" More on Log and Exponential Equations
Equations That Do Not Need Calculators
For exp-equations of t= his type, put all the bases into a common base first.
------=_NextPart_01C957DB.BC238AB0 Content-Location: file:///C:/09764A33/5MoreonLogandExponentialEquations_files/slide0086.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" More on Log and Exponential Equations
Equations That Do Not Need Calculators
For exp-equations of t= his type, put all the bases into a common base first. Then consolidate the exponents on each side
------=_NextPart_01C957DB.BC238AB0 Content-Location: file:///C:/09764A33/5MoreonLogandExponentialEquations_files/slide0087.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" More on Log and Exponential Equations
Equations That Do Not Need Calculators
For exp-equations of t= his type, put all the bases into a common base first. Then consolidate the exponents on each side and put t= he equation into the form
b<= span lang=3DEN-US style=3D'font-size:103%;position:relative;top:-.45em;mso-text= -raise: 30%;mso-fareast-language:ZH-CN'>u =3D bv,
------=_NextPart_01C957DB.BC238AB0 Content-Location: file:///C:/09764A33/5MoreonLogandExponentialEquations_files/slide0088.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" More on Log and Exponential Equations
Equations That Do Not Need Calculators
For exp-equations of t= his type, put all the bases into a common base first. Then consolidate the exponents on each side and put t= he equation into the form
b<= span lang=3DEN-US style=3D'font-size:103%;position:relative;top:-.45em;mso-text= -raise: 30%;mso-fareast-language:ZH-CN'>u =3D bv, then drop the base b= and solve U =3D V.
------=_NextPart_01C957DB.BC238AB0 Content-Location: file:///C:/09764A33/5MoreonLogandExponentialEquations_files/slide0089.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" More on Log and Exponential Equations
Example A:  Solve 2*4= 2x – 1 =3D 81 – 3x
Equations That Do Not Need Calculators
For exp-equations of t= his type, put all the bases into a common base first. Then consolidate the exponents on each side and put t= he equation into the form
b<= span lang=3DEN-US style=3D'font-size:103%;position:relative;top:-.45em;mso-text= -raise: 30%;mso-fareast-language:ZH-CN'>u =3D bv, then drop the base b= and solve U =3D V.
------=_NextPart_01C957DB.BC238AB0 Content-Location: file:///C:/09764A33/5MoreonLogandExponentialEquations_files/slide0090.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" More on Log and Exponential Equations
Example A:  Solve 2*4= 2x – 1 =3D 81 – 3x
Since 4 =3D 22 and 8 =3D 23,
Equations That Do Not Need Calculators
For exp-equations of t= his type, put all the bases into a common base first. Then consolidate the exponents on each side and put t= he equation into the form
b<= span lang=3DEN-US style=3D'font-size:103%;position:relative;top:-.45em;mso-text= -raise: 30%;mso-fareast-language:ZH-CN'>u =3D bv, then drop the base b= and solve U =3D V.
------=_NextPart_01C957DB.BC238AB0 Content-Location: file:///C:/09764A33/5MoreonLogandExponentialEquations_files/slide0091.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" More on Log and Exponential Equations
Example A:  Solve 2*4= 2x – 1 =3D 81 – 3x
Since 4 =3D 22 and 8 =3D 23, put both sides of the equation
in base 2 as            = ;            &n= bsp;            = ;   
Equations That Do Not Need Calculators
For exp-equations of t= his type, put all the bases into a common base first. Then consolidate the exponents on each side and put t= he equation into the form
b<= span lang=3DEN-US style=3D'font-size:103%;position:relative;top:-.45em;mso-text= -raise: 30%;mso-fareast-language:ZH-CN'>u =3D bv, then drop the base b= and solve U =3D V.
------=_NextPart_01C957DB.BC238AB0 Content-Location: file:///C:/09764A33/5MoreonLogandExponentialEquations_files/slide0092.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" More on Log and Exponential Equations
Example A:  Solve 2*4= 2x – 1 =3D 81 – 3x
Since 4 =3D 22 and 8 =3D 23, put both sides of the equation
in base 2 as     = 2*42x – 1 =3D 81 – 3x
        = ;            &n= bsp;               2*(22)= 2x – 1 =3D (23)= 1 – 3x
Equations That Do Not Need Calculators
For exp-equations of t= his type, put all the bases into a common base first. Then consolidate the exponents on each side and put t= he equation into the form
b<= span lang=3DEN-US style=3D'font-size:103%;position:relative;top:-.45em;mso-text= -raise: 30%;mso-fareast-language:ZH-CN'>u =3D bv, then drop the base b= and solve U =3D V.
------=_NextPart_01C957DB.BC238AB0 Content-Location: file:///C:/09764A33/5MoreonLogandExponentialEquations_files/slide0093.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" More on Log and Exponential Equations
Example A:  Solve 2*4= 2x – 1 =3D 81 – 3x
Since 4 =3D 22 and 8 =3D 23, put both sides of the equation
in base 2 as     = 2*42x – 1 =3D 81 – 3x
        = ;            &n= bsp;               2*(22)= 2x – 1 =3D (23)1 – 3x
        = ;            &n= bsp;    2*24x – 2  = =3D 23 – 9x 
Equations That Do Not Need Calculators
For exp-equations of t= his type, put all the bases into a common base first. Then consolidate the exponents on each side and put t= he equation into the form
b<= span lang=3DEN-US style=3D'font-size:103%;position:relative;top:-.45em;mso-text= -raise: 30%;mso-fareast-language:ZH-CN'>u =3D bv, then drop the base b= and solve U =3D V.
------=_NextPart_01C957DB.BC238AB0 Content-Location: file:///C:/09764A33/5MoreonLogandExponentialEquations_files/slide0094.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" More on Log and Exponential Equations
Example A:  Solve 2*4= 2x – 1 =3D 81 – 3x
Since 4 =3D 22 and 8 =3D 23, put both sides of the equation
in base 2 as     = 2*42x – 1 =3D 81 – 3x
        = ;            &n= bsp;               2*(22)= 2x – 1 =3D (23)1 – 3x
        = ;            &n= bsp;    2*24x – 2  = =3D 23 – 9x  
        = ;            &n= bsp;     21 + 4x – 2 =3D 23 – 9x       
Equations That Do Not Need Calculators
For exp-equations of t= his type, put all the bases into a common base first. Then consolidate the exponents on each side and put t= he equation into the form
b<= span lang=3DEN-US style=3D'font-size:103%;position:relative;top:-.45em;mso-text= -raise: 30%;mso-fareast-language:ZH-CN'>u =3D bv, then drop the base b= and solve U =3D V.
------=_NextPart_01C957DB.BC238AB0 Content-Location: file:///C:/09764A33/5MoreonLogandExponentialEquations_files/slide0095.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" More on Log and Exponential Equations
Example A:  Solve 2*4= 2x – 1 =3D 81 – 3x
Since 4 =3D 22 and 8 =3D 23, put both sides of the equation
in base 2 as     = 2*42x – 1 =3D 81 – 3x
        = ;            &n= bsp;               2*(22)= 2x – 1 =3D (23)1 – 3x
        = ;            &n= bsp;    2*24x – 2  = =3D 23 – 9x  
        = ;            &n= bsp;     2=3D 23 – 9x        drop the base 2
        = ;            &n= bsp;   1 + 4x – 2 =3D 3 – 9x
Equations That Do Not Need Calculators
For exp-equations of t= his type, put all the bases into a common base first. Then consolidate the exponents on each side and put t= he equation into the form
b<= span lang=3DEN-US style=3D'font-size:103%;position:relative;top:-.45em;mso-text= -raise: 30%;mso-fareast-language:ZH-CN'>u =3D bv, then drop the base b= and solve U =3D V.
------=_NextPart_01C957DB.BC238AB0 Content-Location: file:///C:/09764A33/5MoreonLogandExponentialEquations_files/slide0096.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" More on Log and Exponential Equations
Example A:  Solve 2*4= 2x – 1 =3D 81 – 3x
Since 4 =3D 22 and 8 =3D 23, put both sides of the equation
in base 2 as     = 2*42x – 1 =3D 81 – 3x
        = ;            &n= bsp;               2*(22)= 2x – 1 =3D (23)1 – 3x
        = ;            &n= bsp;    2*24x – 2  = =3D 23 – 9x  
        = ;            &n= bsp;     2=3D 23 – 9x        drop the base 2
        = ;            &n= bsp;   1 + 4x – 2 =3D 3 – 9x
        = ;            &n= bsp;           4x – 1 =3D 3 – 9x
        = ;            &n= bsp;            = ;      13x =3D 4  à  x =3D 4/13
Equations That Do Not Need Calculators
For exp-equations of t= his type, put all the bases into a common base first. Then consolidate the exponents on each side and put t= he equation into the form
b<= span lang=3DEN-US style=3D'font-size:103%;position:relative;top:-.45em;mso-text= -raise: 30%;mso-fareast-language:ZH-CN'>u =3D bv, then drop the base b= and solve U =3D V.
------=_NextPart_01C957DB.BC238AB0 Content-Location: file:///C:/09764A33/5MoreonLogandExponentialEquations_files/slide0063.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" More on Log and Exponential Equations
For log-equations of t= his type, consolidate the logs on each side first.
Equations That Do Not Need Calculators
------=_NextPart_01C957DB.BC238AB0 Content-Location: file:///C:/09764A33/5MoreonLogandExponentialEquations_files/slide0097.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" More on Log and Exponential Equations
For log-equations of t= his type, consolidate the logs on each side first. There= are two following possibilities.
Equations That Do Not Need Calculators
------=_NextPart_01C957DB.BC238AB0 Content-Location: file:///C:/09764A33/5MoreonLogandExponentialEquations_files/slide0098.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" More on Log and Exponential Equations
For log-equations of t= his type, consolidate the logs on each side first. There= are two following possibilities.
I. If the resulting equation is of the form logb(U) =3D V,
Equations That Do Not Need Calculators
------=_NextPart_01C957DB.BC238AB0 Content-Location: file:///C:/09764A33/5MoreonLogandExponentialEquations_files/slide0099.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" More on Log and Exponential Equations
Equations That Do Not Need Calculators
For log-equations of t= his type, consolidate the logs on each side first. There= are two following possibilities.
I. If the resulting equation is of the form logb(U) =3D V,
drop the = base b by writing it in the exponential form
U =3D eV.
------=_NextPart_01C957DB.BC238AB0 Content-Location: file:///C:/09764A33/5MoreonLogandExponentialEquations_files/slide0100.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" More on Log and Exponential Equations
Example B:  Solve log2(x – 1) + log2(x + 3) =3D 5
Equations That Do Not Need Calculators
For log-equations of t= his type, consolidate the logs on each side first. There= are two following possibilities.
I. If the resulting equation is of the form logb(U) =3D V,
drop the = base b by writing it in the exponential form
U =3D eV.
------=_NextPart_01C957DB.BC238AB0 Content-Location: file:///C:/09764A33/5MoreonLogandExponentialEquations_files/slide0107.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" More on Log and Exponential Equations
Example B:  Solve log2(x – 1) + log2(x + 3) =3D 5
Combine the= log using product rule:   &#= 13;
        = ;            &n= bsp;           log2[(x – 1)(x + 3)] =3D 5
Equations That Do Not Need Calculators
For log-equations of t= his type, consolidate the logs on each side first. There= are two following possibilities.
I. If the resulting equation is of the form logb(U) =3D V,
drop the = base b by writing it in the exponential form
U =3D eV.
------=_NextPart_01C957DB.BC238AB0 Content-Location: file:///C:/09764A33/5MoreonLogandExponentialEquations_files/slide0101.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" More on Log and Exponential Equations
Example B:  Solve log2(x – 1) + log2(x + 3) =3D 5
Combine the= log using product rule:   &#= 13;
        = ;            &n= bsp;           log2[(x – 1)(x + 3)] =3D 5
Write it in exp-form:         (x – 1)(x + 3) =3D exp(5) =3D 25
Equations That Do Not Need Calculators
For log-equations of t= his type, consolidate the logs on each side first. There= are two following possibilities.
I. If the resulting equation is of the form logb(U) =3D V,
drop the = base b by writing it in the exponential form
U =3D eV.
------=_NextPart_01C957DB.BC238AB0 Content-Location: file:///C:/09764A33/5MoreonLogandExponentialEquations_files/slide0102.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" More on Log and Exponential Equations
Example B:  Solve log2(x – 1) + log2(x + 3) =3D 5
Combine the= log using product rule:   &#= 13;
        = ;            &n= bsp;           log2[(x – 1)(x + 3)] =3D 5
Write it in exp-form:         (x – 1)(x + 3) =3D exp(5) =3D 25
        = ;            &n= bsp;            = ;            &n= bsp;       x2 + 2x – 3 =3D 32
Equations That Do Not Need Calculators
For log-equations of t= his type, consolidate the logs on each side first. There= are two following possibilities.
I. If the resulting equation is of the form logb(U) =3D V,
drop the = base b by writing it in the exponential form
U =3D eV.
------=_NextPart_01C957DB.BC238AB0 Content-Location: file:///C:/09764A33/5MoreonLogandExponentialEquations_files/slide0103.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" More on Log and Exponential Equations
Example B:  Solve log2(x – 1) + log2(x + 3) =3D 5
Combine the= log using product rule:   &#= 13;
        = ;            &n= bsp;           log2[(x – 1)(x + 3)] =3D 5
Write it in exp-form:         (x – 1)(x + 3) =3D exp(5) =3D 25
        = ;            &n= bsp;            = ;            &n= bsp;       x2 + 2x – 3 =3D 32 <= /div>
        = ;            &n= bsp;            = ;   x2  + 2x – 35 = =3D 0
Equations That Do Not Need Calculators
For log-equations of t= his type, consolidate the logs on each side first. There= are two following possibilities.
I. If the resulting equation is of the form logb(U) =3D V,
drop the = base b by writing it in the exponential form
U =3D eV.
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Example B:  Solve log2(x – 1) + log2(x + 3) =3D 5
Combine the= log using product rule:   &#= 13;
        = ;            &n= bsp;           log2[(x – 1)(x + 3)] =3D 5
Write it in exp-form:         (x – 1)(x + 3) =3D exp(5) =3D 25
        = ;            &n= bsp;            = ;            &n= bsp;       x2 + 2x – 3 =3D 32 <= /div>
        = ;            &n= bsp;            = ;   x2  + 2x – 35 = =3D 0
        = ;            &n= bsp;            = ;   (x + 7)(x – 5) =3D 0
Equations That Do Not Need Calculators
For log-equations of t= his type, consolidate the logs on each side first. There= are two following possibilities.
I. If the resulting equation is of the form logb(U) =3D V,
drop the = base b by writing it in the exponential form
U =3D eV.
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Example B:  Solve log2(x – 1) + log2(x + 3) =3D 5
Combine the= log using product rule:   &#= 13;
        = ;            &n= bsp;           log2[(x – 1)(x + 3)] =3D 5
Write it in exp-form:         (x – 1)(x + 3) =3D exp(5) =3D 25
        = ;            &n= bsp;            = ;            &n= bsp;       x2 + 2x – 3 =3D 32 <= /div>
        = ;            &n= bsp;            = ;   x2  + 2x – 35 = =3D 0
        = ;            &n= bsp;            = ;   (x + 7)(x – 5) =3D 0
        = ;            &n= bsp;            = ;     x =3D -7, x =3D 5 
Equations That Do Not Need Calculators
For log-equations of t= his type, consolidate the logs on each side first. There= are two following possibilities.
I. If the resulting equation is of the form logb(U) =3D V,
drop the = base b by writing it in the exponential form
U =3D eV.
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Example B:  Solve log2(x – 1) + log2(x + 3) =3D 5
Combine the= log using product rule:   &#= 13;
        = ;            &n= bsp;           log2[(x – 1)(x + 3)] =3D 5
Write it in exp-form:         (x – 1)(x + 3) =3D exp(5) =3D 25
        = ;            &n= bsp;            = ;            &n= bsp;       x2 + 2x – 3 =3D 32 <= /div>
        = ;            &n= bsp;            = ;   x2  + 2x – 35 = =3D 0
        = ;            &n= bsp;            = ;   (x + 7)(x – 5) =3D 0
        = ;            &n= bsp;            = ;     x =3D -7, x =3D 5 
Equations That Do Not Need Calculators
For log-equations of t= his type, consolidate the logs on each side first. There= are two following possibilities.
I. If the resulting equation is of the form logb(U) =3D V,
drop the = base b by writing it in the exponential form
U =3D eV.
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II. If after consolida= ting the logs, the resulting equation is of the for= m logb(U) =3D logb (V),
Equations That Do Not Need Calculators
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II. If after consolida= ting the logs, the resulting equation is of the for= m logb(U) =3D logb (V), just drop the log and s= olve the equation U =3D V.
Equations That Do Not Need Calculators
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Example C:  Solve log4(x + 2) – log4(x – 1) =3D log4 (5)
II. If after consolida= ting the logs, the resulting equation is of the for= m logb(U) =3D logb (V), just drop the log and s= olve the equation U =3D V.
Equations That Do Not Need Calculators
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Example C:  Solve log4(x + 2) – log4(x – 1) =3D log4 (5)
Combine the logs using quotient rule:
  
        = ;            &n= bsp;           log4[        = ;    ] =3D log4(5)<= /div>
(x + 2)
(x – 1)
II. If after consolida= ting the logs, the resulting equation is of the for= m logb(U) =3D logb (V), just drop the log and s= olve the equation U =3D V.
Equations That Do Not Need Calculators
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Example C:  Solve log4(x + 2) – log4(x – 1) =3D log4 (5)
Combine the logs using quotient rule:
  
        = ;            &n= bsp;           log4[        = ;    ] =3D log4(5)
<= /div>
Drop the lo= gs on both sides we get
 
(x + 2)
(x – 1)
(x + 2)
(x – 1)
=3D 5
II. If after consolida= ting the logs, the resulting equation is of the for= m logb(U) =3D logb (V), just drop the log and s= olve the equation U =3D V.
Equations That Do Not Need Calculators
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Example C:  Solve log4(x + 2) – log4(x – 1) =3D log4 (5)
Combine the logs using quotient rule:
  
        = ;            &n= bsp;           log4[        = ;    ] =3D log4(5)
<= /div>
Drop the lo= gs on both sides we get
 
        = ;            &n= bsp;           x + 2 =3D 5(x – 1)    
(x + 2)
(x – 1)
(x + 2)
(x – 1)
=3D 5
II. If after consolida= ting the logs, the resulting equation is of the for= m logb(U) =3D logb (V), just drop the log and s= olve the equation U =3D V.
Equations That Do Not Need Calculators
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Example C:  Solve log4(x + 2) – log4(x – 1) =3D log4 (5)
Combine the logs using quotient rule:
  
        = ;            &n= bsp;           log4[        = ;    ] =3D log4(5)
<= /div>
Drop the lo= gs on both sides we get
 
        = ;            &n= bsp;           x + 2 =3D 5(x – 1)        = ;            &n= bsp;            = ;            &n= bsp;        
        = ;            &n= bsp;            = ;            &n= bsp;  x + 2 =3D 5x – 5
(x + 2)
(x – 1)
(x + 2)
(x – 1)
=3D 5
II. If after consolida= ting the logs, the resulting equation is of the for= m logb(U) =3D logb (V), just drop the log and s= olve the equation U =3D V.
Equations That Do Not Need Calculators
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Example C:  Solve log4(x + 2) – log4(x – 1) =3D log4 (5)
Combine the logs using quotient rule:
  
        = ;            &n= bsp;           log4[        = ;    ] =3D log4(5)
<= /div>
Drop the lo= gs on both sides we get
 
        = ;            &n= bsp;           x + 2 =3D 5(x – 1)        = ;            &n= bsp;            = ;            &n= bsp;        
        = ;            &n= bsp;            = ;            &n= bsp;  x + 2 =3D 5x – 5
        = ;            &n= bsp;            = ;     7 =3D 4x
        = ;            &n= bsp;             7/4 =3D x
(x + 2)
(x – 1)
(x + 2)
(x – 1)
=3D 5
II. If after consolida= ting the logs, the resulting equation is of the for= m logb(U) =3D logb (V), just drop the log and s= olve the equation U =3D V.
Equations That Do Not Need Calculators
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Numerical Equations That Require Calculators
With nume= rical equations we seek both the exact
and appro= ximate answers with calculators.
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Numerical Equations That Require Calculators
With nume= rical equations we seek both the exact
and appro= ximate answers with calculators.
We have s= een some examples in the last section.
------=_NextPart_01C957DB.BC238AB0 Content-Location: file:///C:/09764A33/5MoreonLogandExponentialEquations_files/slide0116.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" More on Log and Exponential Equations
Numerical Equations That Require Calculators
With nume= rical equations we seek both the exact
and appro= ximate answers with calculators.
We have s= een some examples in the last section.
In this s= ection, we tackle the equations that contain
bases tha= t are different from e and 10. 
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Numerical Equations That Require Calculators
With nume= rical equations we seek both the exact
and appro= ximate answers with calculators.
We have s= een some examples in the last section.
In this s= ection, we tackle the equations that contain
bases tha= t are different from e and 10.  We = do this by
changing = the calculation into base e with the
following formula.
------=_NextPart_01C957DB.BC238AB0 Content-Location: file:///C:/09764A33/5MoreonLogandExponentialEquations_files/slide0118.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" More on Log and Exponential Equations
Numerical Equations That Require Calculators
With nume= rical equations we seek both the exact
and appro= ximate answers with calculators.
We have s= een some examples in the last section.
In this s= ection, we tackle the equations that contain
bases tha= t are different from e and 10.  We = do this by
changing = the calculation into base e with the
following formula.
Given x and a base b, we've blog (x) =3D x, 
b
------=_NextPart_01C957DB.BC238AB0 Content-Location: file:///C:/09764A33/5MoreonLogandExponentialEquations_files/slide0119.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" More on Log and Exponential Equations
Numerical Equations That Require Calculators
With nume= rical equations we seek both the exact
and appro= ximate answers with calculators.
We have s= een some examples in the last section.
In this s= ection, we tackle the equations that contain
bases tha= t are different from e and 10.  We = do this by
changing = the calculation into base e with the
following formula.
Given x a= nd a base b, we've blog (x) =3D x, 
apply ln(= ) to both sides:  ln(blog (x)) =3D ln(x)
b
b
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Numerical Equations That Require Calculators
With nume= rical equations we seek both the exact
and appro= ximate answers with calculators.
We have s= een some examples in the last section.
In this s= ection, we tackle the equations that contain
bases tha= t are different from e and 10.  We = do this by
changing = the calculation into base e with the
following formula.
Given x a= nd a base b, we've blog (x) =3D x, 
apply ln(= ) to both sides:  ln(blog (x)) =3D ln(x)
use the p= ower rule     logb(x)ln(b) =3D ln(x)
b
b
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Numerical Equations That Require Calculators
With nume= rical equations we seek both the exact
and appro= ximate answers with calculators.
We have s= een some examples in the last section.
In this s= ection, we tackle the equations that contain
bases tha= t are different from e and 10.  We = do this by
changing = the calculation into base e with the
following formula.
Given x a= nd a base b, we've blog (x) =3D x, 
apply ln(= ) to both sides:  ln(blog (x)) =3D ln(x)
use the p= ower rule     logb(x)ln(b) =3D ln(x)
Therefore= we've the Base-Change Formula
 
        = ;            &n= bsp;    logb(x) =3D
b
b
ln(x)
ln(b)
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Numerical Equations That Require Calculators
To solve equations in bases different from e,
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Numerical Equations That Require Calculators
To solve equations in bases different from e, we
obtain th= e exact solution in the given base first,
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Numerical Equations That Require Calculators
To solve equations in bases different from e, we
obtain th= e exact solution in the given base first, then
use the c= hange of bases formula to obtain the
approxima= te solutions.
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Example D: (Base Change) 
Solve for x if 14 =3D 3·52x+1.
Numerical Equations That Require Calculators
To solve equations in bases different from e, we
obtain th= e exact solution in the given base first, then
use the c= hange of bases formula to obtain the
approxima= te solutions.
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Example D: (Base Change) 
Solve for x if 14 =3D 3·52x+1.
=
Isolate the exp-part:  14/3 =3D 5<= span lang=3DEN-US style=3D'mso-bidi-font-family:Arial;mso-hansi-font-family:Ari= al; font-size:103%;color:#003300;position:relative;top:-.45em;mso-text-raise:3= 0%; mso-fareast-language:ZH-CN'>2x+1
Numerical Equations That Require Calculators
To solve equations in bases different from e, we
obtain th= e exact solution in the given base first, then
use the c= hange of bases formula to obtain the
approxima= te solutions.
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