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Calculation with= Log and Exp
------=_NextPart_01C957DA.A7C5B990 Content-Location: file:///C:/2253A370/3CalculationswithLogandExp_files/slide0011.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.
Calculation with= Log and Exp
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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
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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_01C957DA.A7C5B990 Content-Location: file:///C:/2253A370/3CalculationswithLogandExp_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
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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
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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
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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
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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
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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
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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 alternate 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
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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
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 fo= r 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
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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
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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
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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
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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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document.oncontextmenu=3Dparent._CM; document.onkeydown =3D _KDH; document.ondragstart=3DCancel document.onselectstart=3DCancel self.focus() } function Redirect( frmId ) { var str=3Ddocument.location.hash,idx=3Dstr.indexOf('#'), sId=3DGetSldId() if(idx>=3D0) str=3Dstr.substr(1); if( window.name !=3D frmId && ( sId !=3D str) ) { obj =3D GetObj("Main-File") window.location.href=3Dobj.href+"#"+sId return 1 } return 0 } var MHTMLPrefix =3D CalculateMHTMLPrefix(); function CalculateMHTMLPrefix() { if ( document.location.protocol =3D=3D 'mhtml:') { href=3Dnew String(document.location.href) Start=3Dhref.indexOf('!')+1 End=3Dhref.lastIndexOf('/')+1 if (End < Start) return href.substring(0, Start) else return href.substring(0, End) } return ''; } function GetTags(base,tag) { if(g_supportsPPTHTML) return base.all.tags(tag); else return base.getElementsByTagName(tag); } function UpdNtsPane(){ if(frames["PPTNts"]) PPTNts.location.replace( MHTMLP= refix+GetHrefObj( gCurSld ).mNtsHref ) } function UpdNavPane( sldIndex ){ if(gNavLoaded) PPTNav.UpdNav() } function UpdOtNavPane(){ if(gOtlNavLoaded) PPTOtlNav.UpdOtlNav() } function UpdOtlPane(){ if(gOtlLoaded) PPTOtl.UpdOtl() } function SetHasNts( fVal ) { if( gHasNts !=3D fVal ) { gHasNts=3DfVal UpdNavPane() } } function ToggleOtlText() { gOtlTxtExp=3D!gOtlTxtExp UpdOtlPane() } function ToggleOtlPane() { frmset=3Ddocument.all("PPTHorizAdjust") frm=3Ddocument.all("PPTOtl") if( gOtlOpen ) frmset.cols=3D"*,100%" else frmset.cols=3D"25%,*" gOtlOpen=3D!gOtlOpen frm.noResize=3D!frm.noResize UpdOtNavPane() } function ToggleNtsPane() { frmset=3Ddocument.all("PPTVertAdjust") frm=3Ddocument.all("PPTNts") if( gNtsOpen ) frmset.rows=3D"100%,*" else frmset.rows=3D"*,20%" gNtsOpen=3D!gNtsOpen UpdNtsPane() } function ClearMedia() { if (PPTSld.pptSound) PPTSld.pptSound.loop =3D 0; } function FullScreen() { if ( PPTSld.g_animUseRuntime ) PPTSld.document.body.pause(); ClearMedia(); var href =3D ( document.location.protocol =3D=3D 'mhtml:') ? FULLSCR_HREF = : FULLSCR_HREF+"#"+GetHrefObj(gCurSld).mSldHref; if(PPTNav.event.ctrlKey) { var w =3D (window.screen.availWidth * 1.0) / 2.0 var h =3D w * (PPTSld.g_origH * 1.0) / PPTSld.g_origW win =3D window.open( MHTMLPrefix+href,null,"toolbar=3D0,resizable=3D1,top= =3D0,left=3D0," + "width=3D"+ w + ",height=3D" + h ); if( PPTSld.g_animUseRuntime ) win.document.body.PPTSldFrameset=3Dwindow; } else { win =3D window.open( MHTMLPrefix+href,null,"fullscreen=3Dyes" ); if( PPTSld.g_animUseRuntime ) win.document.body.PPTSldFrameset=3Dwindow; } } function ToggleVNarration() { rObj=3DPPTSld.document.all("NSPlay") if( rObj && !PPTSld.g_animUseRuntime ) { if( (rObj.playState =3D=3D 1)||(rObj.playState =3D=3D 0) ) rObj.Play() else if( rObj.playState =3D=3D 2 ) rObj.Pause() else return; } else if( PPTSld.g_animUseRuntime ) { narObj =3D PPTSld.document.all("narrationID") if( narObj ) narObj.togglePause() } } function GetCurSldNum() { obj=3DGetHrefObj(gCurSld) if( obj.mOrigVis =3D=3D 1 ) return obj.mSldIdx else return gCurSld } function GetNumSlds() { if( GetHrefObj(gCurSld).mOrigVis =3D=3D 1 ) return GetSldList().mNumVisSlds; else return GetSldList().mList.length } function GetSldNum( href ) { for(ii=3D0; ii 1 ) PopSldList(); else if( !IsFramesMode() ) { if( gLoopCont ) GoToFirst() else EndShow() } } function GoToPrevSld() { ii=3DgCurSld-1 if( ii > 0 ) { obj=3DGetHrefObj(ii) while ( obj && ( obj.mVis =3D=3D 0 ) && ( ii>0 ) ) obj=3DGetHrefObj(--ii) if( ii =3D=3D 0 ) ii=3D1 GoToSldNum(ii) } } function GoToFirst(){ GoToSld( GetHrefObj(1).mSldHref ) } function GoToLast() { ii=3DGetSldList().mList.length if( ii !=3D gCurSld ) GoToSld( GetHrefObj(ii).mSldHref ) } function GoToSldNum( num ) { if( PPTSld.event ) PPTSld.event.cancelBubble=3Dtrue obj =3D GetHrefObj( num ) obj.mVis=3D1 gPrevSld=3DgCurSld gCurSld =3D num; PPTSld.location.replace(MHTMLPrefix+obj.mSldHref) if( IsFramesMode() ) { UpdNavPane(); UpdOtlPane(); UpdNtsPane() } } function GoToSld( href ) { if( PPTSld.event ) PPTSld.event.cancelBubble=3Dtrue GetHrefObj( GetSldNum(href) ).mVis=3D1 PPTSld.location.replace(MHTMLPrefix+href) } function SldUpdated( id ) { if( id =3D=3D GetHrefObj(gCurSld).mSldHref ) return gPrevSld=3DgCurSld gCurSld=3DGetSldNum(id) if( IsFramesMode() ) { UpdNavPane(); UpdOtlPane(); UpdNtsPane() } } function PrevSldViewed(){ GoToSld( GetHrefObj(gPrevSld).mSldHref ) } function HasPrevSld() { return ( gIsEndShow || ( gCurSld !=3D 1 && GetHrefO= bj( gCurSld-1 ).mVis =3D=3D 1 )||( GetCurSldNum() > 1 ) ) } function HasNextSld() { return (GetCurSldNum() !=3D GetNumSlds()) } function CloseWindow() { if( HideMenu() ) return; var event =3D PPTSld.event; if( !IsFramesMode() && event && (event.keyCode=3D=3D27 || event.keyCode=3D= =3D32 || event.type=3D=3D"click" ) ) window.close( self ); CatchNumKeys( self, event ); } function Unload() { gIsEndShow=3D0; } function SetupEndShow() { gIsEndShow=3D1; PPTSld.document.body.scroll=3D"no"; PPTSld.document.onkeypress=3DCloseWindow; PPTSld.document.onclick=3DCloseWindow; PPTSld.document.oncontextmenu=3D_CM; } function EndShow() { if( IsFramesMode() ) return if( PPTSld.event ) PPTSld.event.cancelBubble=3Dtrue doc=3DPPTSld.document doc.open() doc.writeln('


' + ENDSHOW_MESG + '

') doc.close() } function SetSldVisited(){ GetSldList().mList[gCurSld-1].mVisited=3Dtrue } function IsSldVisited(){ return GetSldList().mList[gCurSld-1].mVisited } function hrefList( sldHref, visible, advDelay, advClk ) { this.mSldHref=3D this.mNtsHref =3D sldHref this.mOrigVis=3D this.mVis =3D visible this.mVisited=3D false this.mAdvDelay=3D advDelay this.mAdvOnClk=3D advClk } function SldList(arr,curSld,fEnd) { this.mCurSld =3D curSld; this.mList =3D new Array(); var idx =3D 1; for(ii=3D0;ii 0) { PushSldList(sldList,fEnd); gCurSld =3D 1; } else if( PPTSld.event ) PPTSld.event.cancelBubble=3Dtrue } function PushSldList(arr,fEnd) { var ii =3D gSldStack.length; gSldStack[ii] =3D new SldList(arr,gCurSld,fEnd); GoToSld( gSldStack[ii].mList[0].mSldHref ); } function PopSldList() { if (gSldStack[gSldStack.length-1].fEndShow) EndShow() else { gCurSld =3D gSldStack[gSldStack.length-1].mCurSld; gSldStack[gSldStack.length-1] =3D null; gSldStack.length--; var sldList =3D gSldStack[gSldStack.length-1]; GoToSld( sldList.mList[gCurSld - 1].mSldHref ); } } var custShowList=3Dnew Array(); function ImgBtn( oId,bId,w,action ) { var t=3Dthis t.Perform =3D _IBP t.SetActive =3D _IBSetA t.SetInactive=3D _IBSetI t.SetPressed =3D _IBSetP t.SetDisabled=3D _IBSetD t.Enabled =3D _IBSetE t.ChangeIcon =3D null t.UserAction =3D action t.ChgState =3D _IBUI t.mObjId