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In this section, we so= lve simple numerical equations involving log and exponential functions in base 10
or base e.
Calculation with= Log and Exp
------=_NextPart_01C85061.9E78DD30 Content-Location: file:///C:/2253A370/8CalculationwithLogandExp_files/slide0066.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Calculation with Log and Exp
In this section, we so= lve simple numerical equations involving log and exponential functions in base 10
or base e. Most numeri= cal calculations in science are in these two bases.
Calculation with= Log and Exp
------=_NextPart_01C85061.9E78DD30 Content-Location: file:///C:/2253A370/8CalculationwithLogandExp_files/slide0067.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Calculation with Log and Exp
In this section, we so= lve simple numerical equations involving log and exponential functions in base 10
or base e. Most numeri= cal calculations in science are in these two bases. We= need a calculator that has the following functions: ex, 10, ln(x), = and log(x).
Calculation with= Log and Exp
------=_NextPart_01C85061.9E78DD30 Content-Location: file:///C:/2253A370/8CalculationwithLogandExp_files/slide0068.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Calculation with Log and Exp
In this section, we so= lve simple numerical equations involving log and exponential functions in base 10
or base e. Most numeri= cal calculations in science are in these two bases. We= need a calculator that has the following functions: ex, 10, ln(x), = and log(x).
All answers are given = to 3 significant digits. 
Calculation with= Log and Exp
------=_NextPart_01C85061.9E78DD30 Content-Location: file:///C:/2253A370/8CalculationwithLogandExp_files/slide0073.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Calculation with Log and Exp
In this section, we so= lve simple numerical equations involving log and exponential functions in base 10
or base e. Most numeri= cal calculations in science are in these two bases. We= need a calculator that has the following functions: ex, 10, ln(x), = and log(x).
All answers are given = to 3 significant digits. 
6
Example A: Find the answers with a calculator.
a.10        = ;             b. Öe =3D e1/6 <= /div>
  
c. log(4.35)        = ;       d. ln(2/3)
  
Calculation with= Log and Exp
------=_NextPart_01C85061.9E78DD30 Content-Location: file:///C:/2253A370/8CalculationwithLogandExp_files/slide0073_image001.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAS AAEAgAAAAAAAAAIEjI95BQA7 ------=_NextPart_01C85061.9E78DD30 Content-Location: file:///C:/2253A370/8CalculationwithLogandExp_files/slide0074.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Calculation with Log and Exp
In this section, we so= lve simple numerical equations involving log and exponential functions in base 10
or base e. Most numeri= cal calculations in science are in these two bases. We= need a calculator that has the following functions: ex, 10, ln(x), = and log(x).
All answers are given = to 3 significant digits. 
6
Example A: Find the answers with a calculator.
a.10        = ;             b. Öe =3D e1/6 <= /div>
   » 2090        = ;            &n= bsp;    
c. log(4.35)        = ;       d. ln(2/3)
 
Calculation with= Log and Exp
------=_NextPart_01C85061.9E78DD30 Content-Location: file:///C:/2253A370/8CalculationwithLogandExp_files/slide0074_image002.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAS AAEAgAAAAAAAAAIEjI95BQA7 ------=_NextPart_01C85061.9E78DD30 Content-Location: file:///C:/2253A370/8CalculationwithLogandExp_files/slide0075.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Calculation with Log and Exp
In this section, we so= lve simple numerical equations involving log and exponential functions in base 10
or base e. Most numeri= cal calculations in science are in these two bases. We= need a calculator that has the following functions: ex, 10, ln(x), = and log(x).
All answers are given = to 3 significant digits. 
6
Example A: Find the answers with a calculator.
a.10        = ;             b. Öe =3D e1/6 <= /div>
   » 2090        = ;            &n= bsp;    » 1.18
c. log(4.35)        = ;       d. ln(2/3)
 
Calculation with= Log and Exp
------=_NextPart_01C85061.9E78DD30 Content-Location: file:///C:/2253A370/8CalculationwithLogandExp_files/slide0075_image003.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAS AAEAgAAAAAAAAAIEjI95BQA7 ------=_NextPart_01C85061.9E78DD30 Content-Location: file:///C:/2253A370/8CalculationwithLogandExp_files/slide0076.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Calculation with Log and Exp
In this section, we so= lve simple numerical equations involving log and exponential functions in base 10
or base e. Most numeri= cal calculations in science are in these two bases. We= need a calculator that has the following functions: ex, 10, ln(x), = and log(x).
All answers are given = to 3 significant digits. 
6
Example A: Find the answers with a calculator.
a.10        = ;             b. Öe =3D e1/6 <= /div>
   » 2090        = ;            &n= bsp;    » 1.18
c. log(4.35)        = ;       d. ln(2/3)
   »0.638        = ;            &n= bsp;  
Calculation with= Log and Exp
------=_NextPart_01C85061.9E78DD30 Content-Location: file:///C:/2253A370/8CalculationwithLogandExp_files/slide0076_image004.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAS AAEAgAAAAAAAAAIEjI95BQA7 ------=_NextPart_01C85061.9E78DD30 Content-Location: file:///C:/2253A370/8CalculationwithLogandExp_files/slide0077.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Calculation with Log and Exp
In this section, we so= lve simple numerical equations involving log and exponential functions in base 10
or base e. Most numeri= cal calculations in science are in these two bases. We= need a calculator that has the following functions: ex, 10, ln(x), = and log(x).
All answers are given = to 3 significant digits. 
6
Example A: Find the answers with a calculator.
a.10        = ;             b. Öe =3D e1/6 <= /div>
   » 2090        = ;            &n= bsp;    » 1.18
c. log(4.35)        = ;       d. ln(2/3)
   »0.638        = ;            &n= bsp;   » -0.405
Calculation with= Log and Exp
------=_NextPart_01C85061.9E78DD30 Content-Location: file:///C:/2253A370/8CalculationwithLogandExp_files/slide0077_image005.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAS AAEAgAAAAAAAAAIEjI95BQA7 ------=_NextPart_01C85061.9E78DD30 Content-Location: file:///C:/2253A370/8CalculationwithLogandExp_files/slide0078.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Calculation with Log and Exp
In this section, we so= lve simple numerical equations involving log and exponential functions in base 10
or base e. Most numeri= cal calculations in science are in these two bases. We= need a calculator that has the following functions: ex, 10, ln(x), = and log(x).
All answers are given = to 3 significant digits. 
6
Example A: Find the answers with a calculator.
a.10        = ;             b. Öe =3D e1/6 <= /div>
   » 2090        = ;            &n= bsp;    » 1.18
c. log(4.35)        = ;       d. ln(2/3)
   »0.638        = ;            &n= bsp;   » -0.405
These pro= blems may be stated in educational forms.    
Calculation with= Log and Exp
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Example B: Find the x
a. log(x) =3D 3.32        = ;          b. 1/6 =3D ln(x)
   
 
c. 10x =3D 4.35        = ;            &n= bsp; d. 2/3 =3D e
Calculation with= Log and Exp
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Example B: Find the x
a. log(x) =3D 3.32        = ;          b. 1/6 =3D ln(x)
    x =3D103.32 (= » 2090)        = ;      
 
c. 10x =3D 4.35        = ;            &n= bsp; d. 2/3 =3D e
   
Calculation with= Log and Exp
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Example B: Find the x
a. log(x) =3D 3.32        = ;          b. 1/6 =3D ln(x)
    x =3D103.32 (= » 2090)        = ;       e=3D x (» 1.18)= 3;
 
c. 10x =3D 4.35        = ;            &n= bsp; d. 2/3 =3D e
   
Calculation with= Log and Exp
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Example B: Find the x
a. log(x) =3D 3.32        = ;          b. 1/6 =3D ln(x)
    x =3D103.32 (= » 2090)        = ;       e=3D x (» 1.18)= 3;
 
c. 10x =3D 4.35        = ;            &n= bsp; d. 2/3 =3D e
    x =3D log(4.35) (» 0.638)       
Calculation with= Log and Exp
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Example B: Find the x
a. log(x) =3D 3.32        = ;          b. 1/6 =3D ln(x)
    x =3D103.32 (= » 2090)        = ;       e=3D x (» 1.18)= 3;
 
c. 10x =3D 4.35        = ;            &n= bsp; d. 2/3 =3D e
    x =3D log(4.35) (» 0.638)        ln(2/3) =3D x (» -0.405)
Calculation with= Log and Exp
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Example B: Find the x
a. log(x) =3D 3.32        = ;          b. 1/6 =3D ln(x)
    x =3D103.32 (= » 2090)        = ;       e=3D x (» 1.18)= 3;
 
c. 10x =3D 4.35        = ;            &n= bsp; d. 2/3 =3D e
    x =3D log(4.35) (» 0.638)        ln(2/3) =3D x (» -0.405)
An equation is called a log-equations if the unknown is in the log-function as in parts a and b above.  
Calculation with= Log and Exp
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Example B: Find the x
a. log(x) =3D 3.32        = ;          b. 1/6 =3D ln(x)
    x =3D103.32 (= » 2090)        = ;       e=3D x (» 1.18)= 3;
 
c. 10x =3D 4.35        = ;            &n= bsp; d. 2/3 =3D e
    x =3D log(4.35) (» 0.638)        ln(2/3) =3D x (» -0.405)
An equation is called an exponential equations if the unknown i= s in the exponent as in parts c and d.  
An equation is called a log-equations if the unknown is in the log-function as in parts a and b above.  
Calculation with= Log and Exp
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Example B: Find the x
a. log(x) =3D 3.32        = ;          b. 1/6 =3D ln(x)
    x =3D103.32 (= » 2090)        = ;       e=3D x (» 1.18)= 3;
 
c. 10x =3D 4.35        = ;            &n= bsp; d. 2/3 =3D e
    x =3D log(4.35) (» 0.638)        ln(2/3) =3D x (» -0.405)
An equation is called an exponential equations if the unknown i= s in the exponent as in parts c and d.  
An equation is called a log-equations if the unknown is in the log-function as in parts a and b above.  
To solve log-equations, drop the log and write the = problems = in exp-form.
Calculation with= Log and Exp
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Example B: Find the x
a. log(x) =3D 3.32        = ;          b. 1/6 =3D ln(x)
    x =3D103.32 (= » 2090)        = ;       e=3D x (» 1.18)= 3;
 
c. 10x =3D 4.35        = ;            &n= bsp; d. 2/3 =3D e
    x =3D log(4.35) (» 0.638)        ln(2/3) =3D x (» -0.405)
An equation is called an exponential equations if the unknown i= s in the exponent as in parts c and d.  
An equation is called a log-equations if the unknown is in the log-function as in parts a and b above.  
To solve log-equations, drop the log and write the = problems = in exp-form. To solve exponential equations, lower the exponents and write the problems in log-form.<= /span> 
Calculation with= Log and Exp
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More prec= isely, to solve exponential equations,
Calculation with= Log and Exp
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More precisely, to sol= ve exponential equations, we
I. isolate the exponen= tial part that contains the x,
Calculation with= Log and Exp
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More precisely, to sol= ve exponential equations, we
I. isolate the exponen= tial part that contains the x,
II. bring down the exp= onents by writing it in log-form.
Calculation with= Log and Exp
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More precisely, to sol= ve exponential equations, we
I. isolate the exponen= tial part that contains the x,
II. bring down the exp= onents by writing it in log-form.
Example = C: Solve 25 =3D 7*10= 3;
Calculation with= Log and Exp
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More precisely, to sol= ve exponential equations, we
I. isolate the exponen= tial part that contains the x,
II. bring down the exp= onents by writing it in log-form.
Example = C: Solve 25 =3D 7*10
Isolate = the exponential part containing the x,    
25/7 =3D= 102x  = 3;
Calculation with= Log and Exp
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More precisely, to sol= ve exponential equations, we
I. isolate the exponen= tial part that contains the x,
II. bring down the exp= onents by writing it in log-form.
Example = C: Solve 25 =3D 7*10
Isolate = the exponential part containing the x,    
25/7 =3D= 102x  
Bring do= wn the x by restating it in log-form:
log(25/7= ) =3D 2x
Calculation with= Log and Exp
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More precisely, to sol= ve exponential equations, we
I. isolate the exponen= tial part that contains the x,
II. bring down the exp= onents by writing it in log-form.
Example = C: Solve 25 =3D 7*10
Isolate = the exponential part containing the x,    
25/7 =3D= 102x  
Bring do= wn the x by restating it in log-form:
log(25/7= ) =3D 2x
 log(25/7)        
2
=3D x
Calculation with= Log and Exp
------=_NextPart_01C85061.9E78DD30 Content-Location: file:///C:/2253A370/8CalculationwithLogandExp_files/slide0092_image007.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhWwA4AHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAABa ADYAgAAAAAAAAAKjjI+py+0PI5i02ouz3rxv44XiSHpgiaaqdq7uS7bwTGNyjdN3zq97D4wFgkTU r4jMHJNMyrLJfEKR0imxagVis7wtF+f96obiZrj8OqN95HVR7Ta249o5vTuP6Pf8vt35d1cDJxhC WNhxiPgRuOii6GjTGJkCSVlheTmRqcl56UkJGinqSLpoiohaqCrIeudKBxsn60a7ZouGW6Yr5uf7 CxxRAAA7 ------=_NextPart_01C85061.9E78DD30 Content-Location: file:///C:/2253A370/8CalculationwithLogandExp_files/slide0093.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Calculation with Log and Exp
More precisely, to sol= ve exponential equations, we
I. isolate the exponen= tial part that contains the x,
II. bring down the exp= onents by writing it in log-form.
Example = C: Solve 25 =3D 7*10
Isolate = the exponential part containing the x,    
25/7 =3D= 102x  
Bring do= wn the x by restating it in log-form:
log(25/7= ) =3D 2x
 log(25/7)        
2
=3D x » 0.276
Calculation with= Log and Exp
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More precisely, to sol= ve exponential equations, we
I. isolate the exponen= tial part that contains the x,
II. bring down the exp= onents by writing it in log-form.
Example = C: Solve 25 =3D 7*10
Isolate = the exponential part containing the x,    
25/7 =3D= 102x  
Bring do= wn the x by restating it in log-form:
log(25/7= ) =3D 2x
 log(25/7)        
2
=3D x » 0.276
 
Exact answer
Approx. answer
Calculation with= Log and Exp
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Example D: Solve 2.3*e+ 4.1 =3D 12.5
Calculation with= Log and Exp
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Example = D: Solve 2.3*e+ 4.1 =3D 12.5
Isolate = the exp-part: 2.3*e + 4.1 =3D 12.5
Calculation with= Log and Exp
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Example = D: Solve 2.3*e+ 4.1 =3D 12.5
Isolate = the exp-part: 2.3*e + 4.1 =3D 12.5
        = ;            &n= bsp;            = ;      2.3*e =3D 12.5 – 4.1        = ;     
        = ;            &n= bsp;            = ;      2.3*e =3D 8.4
        = ;            &n= bsp;            = ;            
Calculation with= Log and Exp
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Example = D: Solve 2.3*e+ 4.1 =3D 12.5
Isolate = the exp-part: 2.3*e + 4.1 =3D 12.5
        = ;            &n= bsp;            = ;      2.3*e =3D 12.5 – 4.1        = ;     
        = ;            &n= bsp;            = ;      2.3*e =3D 8.4
        = ;            &n= bsp;            = ;             e=3D 8.4/2.3
Calculation with= Log and Exp
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Example = D: Solve 2.3*e+ 4.1 =3D 12.5
Isolate = the exp-part: 2.3*e + 4.1 =3D 12.5
        = ;            &n= bsp;            = ;      2.3*e =3D 12.5 – 4.1        = ;     
        = ;            &n= bsp;            = ;      2.3*e =3D 8.4
        = ;            &n= bsp;            = ;             e=3D 8.4/2.3
Restate = in log-form:        = ;     2 – 3x =3D ln(8.4/2.3)
Calculation with= Log and Exp
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Example = D: Solve 2.3*e+ 4.1 =3D 12.5
Isolate = the exp-part: 2.3*e + 4.1 =3D 12.5
        = ;            &n= bsp;            = ;      2.3*e =3D 12.5 – 4.1        = ;     
        = ;            &n= bsp;            = ;      2.3*e =3D 8.4
        = ;            &n= bsp;            = ;             e=3D 8.4/2.3
Restate = in log-form:        = ;     2 – 3x =3D ln(8.4/2.3)
Solve for x:        = ;            &n= bsp;    2 – ln(8.4/2.3) =3D 3x       
 
Calculation with= Log and Exp
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Example = D: Solve 2.3*e+ 4.1 =3D 12.5
Isolate = the exp-part: 2.3*e + 4.1 =3D 12.5
        = ;            &n= bsp;            = ;      2.3*e =3D 12.5 – 4.1        = ;     
        = ;            &n= bsp;            = ;      2.3*e =3D 8.4
        = ;            &n= bsp;            = ;             e=3D 8.4/2.3
Restate = in log-form:        = ;     2 – 3x =3D ln(8.4/2.3)
Solve for x:        = ;            &n= bsp;    2 – ln(8.4/2.3) =3D 3x       
2-ln(8.4/2.3)        
3
=3D x
 
Calculation with= Log and Exp
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Example = D: Solve 2.3*e+ 4.1 =3D 12.5
Isolate = the exp-part: 2.3*e + 4.1 =3D 12.5
        = ;            &n= bsp;            = ;      2.3*e =3D 12.5 – 4.1        = ;     
        = ;            &n= bsp;            = ;      2.3*e =3D 8.4
        = ;            &n= bsp;            = ;             e=3D 8.4/2.3
Restate = in log-form:        = ;     2 – 3x =3D ln(8.4/2.3)
Solve for x:        = ;            &n= bsp;    2 – ln(8.4/2.3) =3D 3x       
2-ln(8.4/2.3)        
3
=3D x » 0.235
 
Calculation with= Log and Exp
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Example = D: Solve 2.3*e+ 4.1 =3D 12.5
Isolate = the exp-part: 2.3*e + 4.1 =3D 12.5
        = ;            &n= bsp;            = ;      2.3*e =3D 12.5 – 4.1        = ;     
        = ;            &n= bsp;            = ;      2.3*e =3D 8.4
        = ;            &n= bsp;            = ;             e=3D 8.4/2.3
Restate = in log-form:        = ;     2 – 3x =3D ln(8.4/2.3)
Solve for x:        = ;            &n= bsp;    2 – ln(8.4/2.3) =3D 3x       
2-ln(8.4/2.3)        
3
=3D x » 0.235
 
Calculation with= Log and Exp
We solve log-equation in analogous fashion:
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Example = D: Solve 2.3*e+ 4.1 =3D 12.5
Isolate = the exp-part: 2.3*e + 4.1 =3D 12.5
        = ;            &n= bsp;            = ;      2.3*e =3D 12.5 – 4.1        = ;     
        = ;            &n= bsp;            = ;      2.3*e =3D 8.4
        = ;            &n= bsp;            = ;             e=3D 8.4/2.3
Restate = in log-form:        = ;     2 – 3x =3D ln(8.4/2.3)
Solve for x:        = ;            &n= bsp;    2 – ln(8.4/2.3) =3D 3x       
2-ln(8.4/2.3)        
3
=3D x » 0.235
 
Calculation with= Log and Exp
We solve log-equation = in analogous fashion:
I. isolate the log par= t that contains the x,
II. drop the log by wr= iting it in exp-form.
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Calculation with Log and Exp
Example E: Solve 9*log(2x+1)=3D 7
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Calculation with Log and Exp
Example = E: Solve 9*log(2x+1)=3D 7
Isolate = the log-part,   log(2x+1) = =3D 7/9
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Calculation with Log and Exp
Example = E: Solve 9*log(2x+1)=3D 7
Isolate = the log-part,   log(2x+1) =3D 7/9
Write it= in exp-form         2x + 1 =3D 10 
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Calculation with Log and Exp
Example = E: Solve 9*log(2x+1)=3D 7
Isolate = the log-part,   log(2x+1) =3D 7/9
Write it= in exp-form         2x + 1 =3D 10   
Save for x:        = ;            &n= bsp;
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Calculation with Log and Exp
Example = E: Solve 9*log(2x+1)=3D 7
Isolate = the log-part,   log(2x+1) =3D 7/9
Write it= in exp-form         2x + 1 =3D 10   
Save for x:        = ;            &n= bsp; 2x =3D 10 – 1
        = ;            &n= bsp;            = ;       x =3D (10 – 1)/2
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Calculation with Log and Exp
Example = E: Solve 9*log(2x+1)=3D 7
Isolate = the log-part,   log(2x+1) =3D 7/9
Write it= in exp-form         2x + 1 =3D 10   
Save for x:        = ;            &n= bsp; 2x =3D 10 – 1
        = ;            &n= bsp;            = ;       x =3D (10 – 1)/2 » 2.50
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Calculation with Log and Exp
Example = E: Solve 9*log(2x+1)=3D 7
Isolate = the log-part,   log(2x+1) =3D 7/9
Write it= in exp-form         2x + 1 =3D 10   
Save for x:        = ;            &n= bsp; 2x =3D 10 – 1
        = ;            &n= bsp;            = ;       x =3D (10 – 1)/2 » 2.50
Example F: Solve 2.3*log(2–3x)+4.1 =3D 12.5= 3;
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Calculation with Log and Exp
Example = E: Solve 9*log(2x+1)=3D 7
Isolate = the log-part,   log(2x+1) =3D 7/9
Write it= in exp-form         2x + 1 =3D 10   
Save for x:        = ;            &n= bsp; 2x =3D 10 – 1
        = ;            &n= bsp;            = ;       x =3D (10 – 1)/2 » 2.50
Example = F: Solve 2.3*log(2–3x)+4.1 =3D 12.5
2.3*log(= 2–3x) + 4.1 =3D 12.5
------=_NextPart_01C85061.9E78DD30 Content-Location: file:///C:/2253A370/8CalculationwithLogandExp_files/slide0110_image023.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhvQAdAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQC8 ABsAgAAAAAAAAAKIjI+py+0Po5y0Wguy3rz7D4biSJbmiaamobbuC8fy3LH0jef6/tn8DwwKQb6h 8YiUFZPMprMXeEqnziX1itVZs9yua+sNi0XgsfkMKKPXXTX7TXXD5005/W604/c/Pf+P4wc4GCNI eKhiiLhYosj4SBQFOfniSHlpeTmZqfl48QkaKjpKWtpQAAA7 ------=_NextPart_01C85061.9E78DD30 Content-Location: file:///C:/2253A370/8CalculationwithLogandExp_files/slide0111.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Calculation with Log and Exp
Calculation with Log and Exp
Example = E: Solve 9*log(2x+1)=3D 7
Isolate = the log-part,   log(2x+1) =3D 7/9
Write it= in exp-form         2x + 1 =3D 10   
Save for x:        = ;            &n= bsp; 2x =3D 10 – 1
        = ;            &n= bsp;            = ;       x =3D (10 – 1)/2 » 2.50
Example = F: Solve 2.3*log(2–3x)+4.1 =3D 12.5
2.3*log(= 2–3x) + 4.1 =3D 12.5
2.3*log(= 2–3x) =3D 12.5 – 4.1
2.3*log(= 2–3x) =3D 8.4
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Calculation with Log and Exp
Example = E: Solve 9*log(2x+1)=3D 7
Isolate = the log-part,   log(2x+1) =3D 7/9
Write it= in exp-form         2x + 1 =3D 10   
Save for x:        = ;            &n= bsp; 2x =3D 10 – 1
        = ;            &n= bsp;            = ;       x =3D (10 – 1)/2 » 2.50
Example = F: Solve 2.3*log(2–3x)+4.1 =3D 12.5
2.3*log(= 2–3x) + 4.1 =3D 12.5
2.3*log(= 2–3x) =3D 12.5 – 4.1
2.3*log(= 2–3x) =3D 8.4
log(2 &#= 8211; 3x) =3D 8.4/2.3
    
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Calculation with Log and Exp
Example = E: Solve 9*log(2x+1)=3D 7
Isolate = the log-part,   log(2x+1) =3D 7/9
Write it= in exp-form         2x + 1 =3D 10   
Save for x:        = ;            &n= bsp; 2x =3D 10 – 1
        = ;            &n= bsp;            = ;       x =3D (10 – 1)/2 » 2.50
Example = F: Solve 2.3*log(2–3x)+4.1 =3D 12.5
2.3*log(= 2–3x) + 4.1 =3D 12.5
2.3*log(= 2–3x) =3D 12.5 – 4.1
2.3*log(= 2–3x) =3D 8.4
log(2 &#= 8211; 3x) =3D 8.4/2.3
      2 –= 3x =3D 108.4/2.3
------=_NextPart_01C85061.9E78DD30 Content-Location: file:///C:/2253A370/8CalculationwithLogandExp_files/slide0113_image026.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhvQAdAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQC8 ABsAgAAAAAAAAAKIjI+py+0Po5y0Wguy3rz7D4biSJbmiaamobbuC8fy3LH0jef6/tn8DwwKQb6h 8YiUFZPMprMXeEqnziX1itVZs9yua+sNi0XgsfkMKKPXXTX7TXXD5005/W604/c/Pf+P4wc4GCNI eKhiiLhYosj4SBQFOfniSHlpeTmZqfl48QkaKjpKWtpQAAA7 ------=_NextPart_01C85061.9E78DD30 Content-Location: file:///C:/2253A370/8CalculationwithLogandExp_files/slide0114.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Calculation with Log and Exp
Calculation with Log and Exp
Example = E: Solve 9*log(2x+1)=3D 7
Isolate = the log-part,   log(2x+1) =3D 7/9
Write it= in exp-form         2x + 1 =3D 10   
Save for x:        = ;            &n= bsp; 2x =3D 10 – 1
        = ;            &n= bsp;            = ;       x =3D (10 – 1)/2 » 2.50
Example = F: Solve 2.3*log(2–3x)+4.1 =3D 12.5
2.3*log(= 2–3x) + 4.1 =3D 12.5
2.3*log(= 2–3x) =3D 12.5 – 4.1
2.3*log(= 2–3x) =3D 8.4
log(2 &#= 8211; 3x) =3D 8.4/2.3
      2 –= 3x =3D 108.4/2.3
      2 –= 108.4/2.3 =3D 3x
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Calculation with Log and Exp
Example = E: Solve 9*log(2x+1)=3D 7
Isolate = the log-part,   log(2x+1) =3D 7/9
Write it= in exp-form         2x + 1 =3D 10   
Save for x:        = ;            &n= bsp; 2x =3D 10 – 1
        = ;            &n= bsp;            = ;       x =3D (10 – 1)/2 » 2.50
Example = F: Solve 2.3*log(2–3x)+4.1 =3D 12.5
2.3*log(= 2–3x) + 4.1 =3D 12.5
2.3*log(= 2–3x) =3D 12.5 – 4.1
2.3*log(= 2–3x) =3D 8.4
log(2 &#= 8211; 3x) =3D 8.4/2.3
      2 –= 3x =3D 108.4/2.3
      2 –= 108.4/2.3 =3D 3x
      2 –= 108.4/2.3   
=3D x » -1495
3
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Solve the following exponential equations, give the exact and the approximate solutions.
1.   5e2x = =3D 7        = ;            &n= bsp;            = ;            &n= bsp;   2.   3e - 2x+1 =3D 6        = ;   
Exact answ= er: x =3D ½* LN(7/5)        = ;           Exact answer: x =3D (1 – LN(2)) /2        = ;           
Aprox&= iacute;mate:  0.168        = ;            &n= bsp;           Aproxímate:  0.153        = ;            &n= bsp;     
3.   4 – e 3x+ 1 =3D 2        = ;            &n= bsp;            = ;        4.  2* 10 3x - 2 =3D 5
Exact answ= er: x =3D (LN(2) – 1)/3        = ;        Exact answer: x =3D (LOG(5/2) + 2)/3        = ;            &n= bsp;       
Approximat= e: - 0.102        = ;            &n= bsp;           Approximate:  0.799        = ;            &n= bsp;     
5.   6 + 3* 10 1- x =3D 10        = ;            &n= bsp;           6.  -7 – 3*10 2x - 1 =3D -24
Exact answ= er:  x =3D 1 – LOG(4/3)        = ;      Exact answer: x =3D (LOG(17/3)+1)/2        = ;            &n= bsp;      
Aprox&= iacute;mate:  0.875        = ;            &n= bsp;           Aproxímate:   0.877        = ;            &n= bsp;    
7.   8 =3D 12 – 2e = 2- x         = ;            &n= bsp;            = ;   8. 5*10 2 - 3x + 3 = =3D 14
Exact answ= er: x =3D 2 – LN(2)        = ;             Exact answer: x =3D (2 – LOG(11/5)) /3        = ;            &n= bsp;     
Approximat= e:   1.31        = ;            &n= bsp;            Approximate:  0.553        = ;            &n= bsp;     
Solve the following log equations, give the exact and the approximate solutions. = ;
9.  LOG(3x+1) =3D 3/5        = ;            &n= bsp;            10.  LN(2 – x) = =3D -2/3
Exact answ= er: x =3D (103/5 – 1)/3        = ;         Exact answer: x =3D 2 – e -2/3        = ;            &n= bsp;           
Approximat= e:  0.994        = ;            &n= bsp;            Approximate: 1.49        = ;            &n= bsp;       
11.   2LOG(2x –3) =3D 1/3        = ;            &n= bsp;      12.  2 + Log(4 –= 2x) =3D -8
Exact answ= er: x =3D (101/6 + 3)/2        = ;         Exact answer: x =3D (4 – 10-10)/2        = ;            &n= bsp;            = ;          
Approximat= e:  2.23        = ;            &n= bsp;            = ;  Approximate: 2.000        = ;            &n= bsp;       
13.  3 – 5LN(3x +1) =3D -8        = ;            &n= bsp;     14.  -3 +5LOG(1 –= ; 2x) =3D 9
Exact answ= er:  x =3D (e11/5 – 1 )/3        = ;        Exact answer:  x =3D (= 1 – 10 12/5)/2        = ;            &n= bsp;           
Approximat= e:  2.68        = ;            &n= bsp;            = ;  Approximate:  -125        = ;            &n= bsp;     
15.  2LN(2x – 1) – 3 =3D 5=         = ;            &n= bsp;     16.  7 – 2LN(12x= +15) =3D23   
Exact answ= er: x =3D (e4+1)/2        = ;            &n= bsp;  Exact answer: x =3D (e – 15 )/12        = ;            &n= bsp;          
Approximat= e:  27.8        = ;            &n= bsp;            = ;  Approximate:  -1.25     
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parent.IsFullScrMode() ) ) document.body.scroll=3D"no" _RSW() if( IsWin("PPTSld") && parent.IsFullScrMode() ) FullScrInit(); =09 MakeSldVis(); ChkAutoAdv() if( runAnimations ) { if( document.all("NSPlay") ) document.all("NSPlay").autoStart =3D false; if( sld.filters && sld.filters.revealtrans ) setTimeout( "document.body.start()", sld.filters.revealtrans.duration * = 1000 ); else document.body.start(); } } function MakeSldVis()=20 { var fTrans=3Dg_showAnimation && SldHasTrans() if( fTrans )=09 { if( g_bgSound ) { idx=3Dg_bgSound.indexOf(","); pptSound.src=3Dg_bgSound.substr( 0, idx ); pptSound.loop=3D -(parseInt(g_bgSound.substr(idx+1))); } SlideObj.filters.revealtrans.Apply()=09 } SlideObj.style.visibility=3D"visible" if( fTrans ) SlideObj.filters.revealtrans.Play() } function MakeNotesVis()=20 { if( !IsNts() ) return false=20 SlideObj.style.display=3D"none" nObj =3D document.all.item("NotesObj") parent.SetHasNts(0) if( nObj ) {=20 nObj.style.display=3D"" parent.SetHasNts(1) } return 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