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------=_NextPart_01C97CE4.DD467620 Content-Location: file:///C:/AEC9468E/9DerivativeofLogandExponentialFunctions._files/master03.xml Content-Transfer-Encoding: quoted-printable Content-Type: text/xml; charset="utf-8" ------=_NextPart_01C97CE4.DD467620 Content-Location: file:///C:/AEC9468E/9DerivativeofLogandExponentialFunctions._files/preview.wmf Content-Transfer-Encoding: base64 Content-Type: image/x-wmf AQAJAAAD8gAAAAUASQAAAAAABAAAAAMBCAAFAAAACwIAAAAABQAAAAwC7wLpAwMAAAAeAAcAAAD8 AgAA////AAAABAAAAC0BAAAIAAAA+gIFAAAAAAD///8ABAAAAC0BAQAMAAAAQAkhAPAAAAAAAAAA 7wLpA/////8IAAAA+gIAAAAAAAAAAAAABAAAAC0BAgAEAAAALQEAAAQAAAAnAf//HAAAAPsC1P8A AAAAAACQAQAAAAAAQAAiQXJpYWwAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAEAAAALQEDAAQA AAAuARgABAAAAAIBAQAFAAAACQIAAAACSQAAADIKOQBCACwAAABEZXJpdmF0aXZlcyBvZiBMb2cg YW5kIEV4cG9uZW50aWFsIEZ1bmN0aW9ucyAAGAAPAAoAFwAYAAwACgAXABgAFgANABgADQAMABkA GAAZAAwAGQAYABkADQAdABYAGQAZABgAGQAYAAwACwAYAAoADAAcABgAGQAWAAwACwAYABgAFgAE AAAALgEAABwAAAD7AhAABwAAAAAAvAIAAAAAAQICIlN5c3RlbQAAAAAAABgAAABoaRMAAQAAAOQE AAAAAAAABAAAAC0BBAAEAAAA8AEDAAMAAAAAAA== ------=_NextPart_01C97CE4.DD467620 Content-Location: file:///C:/AEC9468E/9DerivativeofLogandExponentialFunctions._files/pres.xml Content-Transfer-Encoding: quoted-printable Content-Type: text/xml; charset="utf-8" ------=_NextPart_01C97CE4.DD467620 Content-Location: file:///C:/AEC9468E/9DerivativeofLogandExponentialFunctions._files/slide0004.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Deri= vatives of Log and Exponential Functions
------=_NextPart_01C97CE4.DD467620 Content-Location: file:///C:/AEC9468E/9DerivativeofLogandExponentialFunctions._files/slide0014.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
We may restate the relation
“x is base-b to the power p”<= span style=3D'mso-spacerun:yes'> 
to be the same as
“p is base-b log of x” 
&#= 13;
(x=3Db= p)&#= 13;
(p =3D logb(x))
Deri= vatives of Log and Exponential Functions
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We may restate the relation
“x is base-b to the power p”<= span style=3D'mso-spacerun:yes'> 
to be the same as
“p is base-b log of x” 
&#= 13;
“bp=3Dx” is called the exponential form,
“logb(x) =3D p” is called the log form of the relation.=  
(x=3Db= p)&#= 13;
(p =3D logb(x))
Deri= vatives of Log and Exponential Functions
------=_NextPart_01C97CE4.DD467620 Content-Location: file:///C:/AEC9468E/9DerivativeofLogandExponentialFunctions._files/slide0016.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
We may restate the relation
“x is base-b to the power p”<= span style=3D'mso-spacerun:yes'> 
to be the same as
“p is base-b log of x” 
&#= 13;
“bp=3Dx” is called the exponential form,
“logb(x) =3D p” is called the log form of the relation.=  
(x=3Db= p)&#= 13;
(p =3D logb(x))
Deri= vatives of Log and Exponential Functions
Log(x) is denoted as L= n(x). (The natural log.)
= ;
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We may restate the relation
“x is base-b to the power p”<= span style=3D'mso-spacerun:yes'> 
to be the same as
“p is base-b log of x” 
&#= 13;
“bp=3Dx” is called the exponential form,
“logb(x) =3D p” is called the log form of the relation.=  
(x=3Db= p)&#= 13;
(p =3D logb(x))
Deri= vatives of Log and Exponential Functions
Log(x) is denoted as L= n(x). (The natural log.)
The range of bx  is always positive.
------=_NextPart_01C97CE4.DD467620 Content-Location: file:///C:/AEC9468E/9DerivativeofLogandExponentialFunctions._files/slide0018.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
We may restate the relation
“x is base-b to the power p”<= span style=3D'mso-spacerun:yes'> 
to be the same as
“p is base-b log of x” 
&#= 13;
“bp=3Dx” is called the exponential form,
“logb(x) =3D p” is called the log form of the relation.=  
(x=3Db= p)&#= 13;
(p =3D logb(x))
Deri= vatives of Log and Exponential Functions
Log(x) is denoted as L= n(x). (The natural log.)
The range of bx  is always positive.
The domain of logb(x) is the set of all x>0.
------=_NextPart_01C97CE4.DD467620 Content-Location: file:///C:/AEC9468E/9DerivativeofLogandExponentialFunctions._files/slide0005.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Example:
a. logb(1) =3D 0        = ;       b0 =3D 1        = ;  
b. logb(bp) =3D p        = ;      bp =3D bp
c. 82/3  =3D 4        = ;           log8(4) =3D 2/3
d. A/P =3D ert        = ;          Ln= (A/P) =3D rt
Deri= vatives of Log and Exponential Functions
------=_NextPart_01C97CE4.DD467620 Content-Location: file:///C:/AEC9468E/9DerivativeofLogandExponentialFunctions._files/slide0005_image001.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhNwCrAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAMAAgAw AKQAgAAAAAAAAAL/hI+pu8GfgoMUyFnRxbntvkndE4pkiaaqqqzu6xrwTJdWjb9azn9eOGKkKsBg qwgyGXdKI2d5g0qn1Kr1is1qt9yu9wsOi8fksvmMTqvX7Lb7DY/L5/S6/Y7P6/f8vv8/0gQlmETo dBGIJGUDMUTlGNEjiTFZKWJZeYCZ86M4iELBOCVKhPhoelhlCMja6voKGys7S1tre4ubq7vL2+v7 Cxz8qfpUWHy42plsvHxD+vkss1mjOc3jbE0tna1zFI3s6Y26GH7SXHruQSzM3u7+Dh8vP09fb3+P n6+/zz94Snws1Dho6bAVNOdDCKhTC5lwi2HwoQ6JMxxSLBcxYKJvDZEGLuGoUKOxResgFAAAOw== ------=_NextPart_01C97CE4.DD467620 Content-Location: file:///C:/AEC9468E/9DerivativeofLogandExponentialFunctions._files/slide0005_image002.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhlQDBAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAACU AL8AgQAAAAAAALvg4wECAwL/jI6py+0Powig2ouz3rx7YEjiSJYI9aXq+oXmC8MoS9eqG+f6M9v+ f8HthsQe8FgTEpcxI/KZUjKnJCf0upFStxAr9mvRcscLL/grJqsn53ZwDS+75+n41jxH1u1TfB64 x7fk9+cTKDhEWJiECKe4yHLYmPMIeTOpVmnZgkmmudkh2WnyCZo1ylVqmiGKOqK6+ubaFws4S1tr eMsEm9u6y5OrC5wobPNLzNBbi5yssBzb7HxizDjdVE0jfQ29uj3dbfrtHA46nly+eU6cbrkO3A75 vhu/OH9bXxiwz9/v/w8woMCBBAvyy4YwocKFDBs6fAgxosSJFCtavIgxo0Zm/wY7evwIst/Fe7Py /SHpymQelKhU0rmGzSLLUS7dzOxUs81NTDnP7JzUE8zPRkHRwJQx8uiLoliGImJ6xakgqFCk8qH6 xKodrHqUkkrqtQrYsCK4HtEax6wtsmXHsu3i9q0DtT/QOoorVxnevM/28qUm8+9cv4LpDhMsJzDi xBXtrjF8bDFjio4zEf4L2ZpkwI03J8iszTNnyqIPgI5Umo3MkKxbu963Mbbs2bRr276NO7fu3bx7 +979Orjw4cSDy0qttEdl5CWUM09+/Dm46NLRUa8O7zp2fNq3p+zunSb48DzHkydq/vzU9Oq3sm9/ Nwz87/Lni69vvzz+/Oj38+5f799/7gUoYHwVLJefcwUCeOCCDILg4FXvRVgMgRTeMeGFlGSoIVIW djgIhyA2J+KIr5RoogQKpohLgyy2COGLvKAoYwMr1rjDjThuGEZxPv4I5I+/DUlkkUYeiWSSSi7J ZJNOPglllFJOSVuQVl6JJUgx7jgIggLuwyUvXv4HZphFjMlfmWbmiGaCbdqn5pqUvDlfnHLKQCd8 dt5JSp7t7clnFX6qB2igZQ16XqGGRqDoojwgSl6jjtoIaXiSTlpGpd5diulnmm7HaaemfYpdqKKa 2imqmKo6KauOuroorIbKGiitfGaJa6668lMAADs= ------=_NextPart_01C97CE4.DD467620 Content-Location: file:///C:/AEC9468E/9DerivativeofLogandExponentialFunctions._files/slide0019.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Example:
a. logb(1) =3D 0        = ;       b0 =3D 1        = ;  
b. logb(bp) =3D p        = ;      bp =3D bp
c. 82/3  =3D 4        = ;           log8(4) =3D 2/3
d. A/P =3D ert        = ;          Ln= (A/P) =3D rt
Deri= vatives of Log and Exponential Functions
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Example:
a. logb(1) =3D 0        = ;       b0 =3D 1        = ;  
b. logb(bp) =3D p        = ;      bp =3D bp
c. 82/3  =3D 4        = ;           log8(4) =3D 2/3
d. A/P =3D ert        = ;          Ln= (A/P) =3D rt
Deri= vatives of Log and Exponential Functions
------=_NextPart_01C97CE4.DD467620 Content-Location: file:///C:/AEC9468E/9DerivativeofLogandExponentialFunctions._files/slide0020_image005.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhNwCrAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAMAAgAw AKQAgAAAAAAAAAL/hI+pu8GfgoMUyFnRxbntvkndE4pkiaaqqqzu6xrwTJdWjb9azn9eOGKkKsBg qwgyGXdKI2d5g0qn1Kr1is1qt9yu9wsOi8fksvmMTqvX7Lb7DY/L5/S6/Y7P6/f8vv8/0gQlmETo dBGIJGUDMUTlGNEjiTFZKWJZeYCZ86M4iELBOCVKhPhoelhlCMja6voKGys7S1tre4ubq7vL2+v7 Cxz8qfpUWHy42plsvHxD+vkss1mjOc3jbE0tna1zFI3s6Y26GH7SXHruQSzM3u7+Dh8vP09fb3+P n6+/zz94Snws1Dho6bAVNOdDCKhTC5lwi2HwoQ6JMxxSLBcxYKJvDZEGLuGoUKOxResgFAAAOw== ------=_NextPart_01C97CE4.DD467620 Content-Location: file:///C:/AEC9468E/9DerivativeofLogandExponentialFunctions._files/slide0020_image006.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhlQBgAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAACU AF4AgQAAAAAAALvg4wECAwL/hB2py+0Po5y02usOCKL7D4biSJbmiabqGgYax8byTNe24B7wzff+ X8ohgMSicSbcHJfM5ie5c0qnPCj1ipVZs9wuaesNh8HiMpZsTjvR6raR7Y774PJ6jW7Ps/D6/onv FygCKFjYQWgoiJjot8io5/hoFykpR1npdompprlp1ukpBhrqNUrKZXp69qKqyNra+AoLKTs7WWtr iZubucvL6fv7GSws+oKBnKy8zNysoAEdLT1NXW19jZ2tvc3d7f0NHi4+Tl5ufo6err7O3u7+Dh8v P09fb3+Pn6+/z9/v/w8woMCBBAsavOYsocKFDDEoKTYpCsRMEidyqmjxE8aMeKI2cizl8SOqkCLP kCxJJQFKjSvLqGw55iRMJi9njrTZpSbOKzp3Tunpc43MoD+AEl1i9OiboUpvJG0K5CnUOUynIqlq NYbUrDa2cqXh9asWrGJRhC274izaFGrX/iHrdkTbuF/g0gUx924Lu3oP8e3bMLDgwQsLAAA7 ------=_NextPart_01C97CE4.DD467620 Content-Location: file:///C:/AEC9468E/9DerivativeofLogandExponentialFunctions._files/slide0021.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Example:
a. logb(1) =3D 0        = ;       b0 =3D 1        = ;  
b. logb(bp) =3D p        = ;      bp =3D bp
c. 82/3  =3D 4        = ;           log8(4) =3D 2/3
d. A/P =3D ert        = ;          Ln= (A/P) =3D rt
Deri= vatives of Log and Exponential Functions
------=_NextPart_01C97CE4.DD467620 Content-Location: file:///C:/AEC9468E/9DerivativeofLogandExponentialFunctions._files/slide0021_image007.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhNwCrAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAMAAgAw AKQAgAAAAAAAAAL/hI+pu8GfgoMUyFnRxbntvkndE4pkiaaqqqzu6xrwTJdWjb9azn9eOGKkKsBg qwgyGXdKI2d5g0qn1Kr1is1qt9yu9wsOi8fksvmMTqvX7Lb7DY/L5/S6/Y7P6/f8vv8/0gQlmETo dBGIJGUDMUTlGNEjiTFZKWJZeYCZ86M4iELBOCVKhPhoelhlCMja6voKGys7S1tre4ubq7vL2+v7 Cxz8qfpUWHy42plsvHxD+vkss1mjOc3jbE0tna1zFI3s6Y26GH7SXHruQSzM3u7+Dh8vP09fb3+P n6+/zz94Snws1Dho6bAVNOdDCKhTC5lwi2HwoQ6JMxxSLBcxYKJvDZEGLuGoUKOxResgFAAAOw== ------=_NextPart_01C97CE4.DD467620 Content-Location: file:///C:/AEC9468E/9DerivativeofLogandExponentialFunctions._files/slide0021_image008.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhlQAqAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAACU ACkAgQAAAAAAALvg4wECAwKmjI+py+0Po5z0gSCy3rz7D4biSJbmGRroyrbuC3dqTNf2bc74zvev 7gsKhxwg8Yi8GZPM5mrpjEo/0Kl1Wr1qmdmtd9j9innhsblWPqtd6bX71H7LRfG5XYa56+H5vZ/e 9yeIN1hYFGg4WJeotsho5vgoFinpRVmpdYlppbkp1enpBBrKhUjqNnp6lKoKZtoK+Qo7KTtrWWub iZvLWeH7CxwsPMxQAAA7 ------=_NextPart_01C97CE4.DD467620 Content-Location: file:///C:/AEC9468E/9DerivativeofLogandExponentialFunctions._files/slide0022.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Example:
a. logb(1) =3D 0        = ;       b0 =3D 1        = ;  
b. logb(bp) =3D p        = ;      bp =3D bp
c. 82/3  =3D 4        = ;           log8(4) =3D 2/3
d. A/P =3D ert        = ;          Ln= (A/P) =3D rt
Deri= vatives of Log and Exponential Functions
------=_NextPart_01C97CE4.DD467620 Content-Location: file:///C:/AEC9468E/9DerivativeofLogandExponentialFunctions._files/slide0022_image009.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhNwCrAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAMAAgAw AKQAgAAAAAAAAAL/hI+pu8GfgoMUyFnRxbntvkndE4pkiaaqqqzu6xrwTJdWjb9azn9eOGKkKsBg qwgyGXdKI2d5g0qn1Kr1is1qt9yu9wsOi8fksvmMTqvX7Lb7DY/L5/S6/Y7P6/f8vv8/0gQlmETo dBGIJGUDMUTlGNEjiTFZKWJZeYCZ86M4iELBOCVKhPhoelhlCMja6voKGys7S1tre4ubq7vL2+v7 Cxz8qfpUWHy42plsvHxD+vkss1mjOc3jbE0tna1zFI3s6Y26GH7SXHruQSzM3u7+Dh8vP09fb3+P n6+/zz94Snws1Dho6bAVNOdDCKhTC5lwi2HwoQ6JMxxSLBcxYKJvDZEGLuGoUKOxResgFAAAOw== ------=_NextPart_01C97CE4.DD467620 Content-Location: file:///C:/AEC9468E/9DerivativeofLogandExponentialFunctions._files/slide0023.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Example:
a. logb(1) =3D 0        = ;       b0 =3D 1        = ;  
b. logb(bp) =3D p        = ;      bp =3D bp
c. 82/3  =3D 4        = ;           log8(4) =3D 2/3
d. A/P =3D ert        = ;          Ln= (A/P) =3D rt
The log. and exp. functions are the inverse of &#= 13;
each other
Deri= vatives of Log and Exponential Functions
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Example:
a. logb(1) =3D 0        = ;       b0 =3D 1        = ;  
b. logb(bp) =3D p        = ;      bp =3D bp
c. 82/3  =3D 4        = ;           log8(4) =3D 2/3
d. A/P =3D ert        = ;          Ln= (A/P) =3D rt
The log. and exp. functions are the inverse of &#= 13;
each other, hence (stated in base e) <= /div>
Ln(eu) = =3D u
Deri= vatives of Log and Exponential Functions
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Example:
a. logb(1) =3D 0        = ;       b0 =3D 1        = ;  
b. logb(bp) =3D p        = ;      bp =3D bp
c. 82/3  =3D 4        = ;           log8(4) =3D 2/3
d. A/P =3D ert        = ;          Ln= (A/P) =3D rt
The log. and exp. functions are the inverse of &#= 13;
each other, hence (stated in base e) <= /div>
Ln(eu) = =3D u  and  eLn(u) =3D= u.
Deri= vatives of Log and Exponential Functions
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Example: Simplify
a. Ln(      ) =3D        = ;  
e2  
Öe 
Deri= vatives of Log and Exponential Functions
------=_NextPart_01C97CE4.DD467620 Content-Location: file:///C:/AEC9468E/9DerivativeofLogandExponentialFunctions._files/slide0006_image013.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhJwAEAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAl AAMAgAAAAAAAAAIOjI+pywEPo5y0WtjUBQUAOw== ------=_NextPart_01C97CE4.DD467620 Content-Location: file:///C:/AEC9468E/9DerivativeofLogandExponentialFunctions._files/slide0026.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Example: Simplify
a. Ln(      ) =3D        = ;  
e2  
Öe 
e2  
 e1/2<= span style=3D'font-size:178%;color:#006600'>&n= bsp;
Ln(      ) =
Deri= vatives of Log and Exponential Functions
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Example: Simplify
a. Ln(      ) =3D        = ;  
e2  
Öe 
e2  
 e1/2<= span style=3D'font-size:178%;color:#006600'>&n= bsp;
Ln(      ) =3D Ln(= e3/2)
Deri= vatives of Log and Exponential Functions
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Example: Simplify
a. Ln(      ) =3D        = ;  
e2  
Öe 
e2  
 e1/2<= span style=3D'font-size:178%;color:#006600'>&n= bsp;
Ln(      ) =3D Ln(= e3/2) =3D 3/2
Deri= vatives of Log and Exponential Functions
------=_NextPart_01C97CE4.DD467620 Content-Location: file:///C:/AEC9468E/9DerivativeofLogandExponentialFunctions._files/slide0028_image018.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhJwAEAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAl AAMAgAAAAAAAAAIOjI+pywEPo5y0WtjUBQUAOw== ------=_NextPart_01C97CE4.DD467620 Content-Location: file:///C:/AEC9468E/9DerivativeofLogandExponentialFunctions._files/slide0028_image019.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhJgACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAm AAEAgAAAAAAAAAIFjI+py1YAO1== ------=_NextPart_01C97CE4.DD467620 Content-Location: file:///C:/AEC9468E/9DerivativeofLogandExponentialFunctions._files/slide0029.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Example: Simplify
a. Ln(      ) =3D        = ;  
e2  
Öe 
e2  
 e1/2<= span style=3D'font-size:178%;color:#006600'>&n= bsp;
Ln(      ) =3D Ln(= e3/2) =3D 3/2
b. e2Ln(x)+1        
Deri= vatives of Log and Exponential Functions
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Example: Simplify
a. Ln(      ) =3D        = ;  
e2  
Öe 
e2  
 e1/2<= span style=3D'font-size:178%;color:#006600'>&n= bsp;
Ln(      ) =3D Ln(= e3/2) =3D 3/2
b. e2Ln(x)+1 =3D e2Ln(x) * e        = ; 
Deri= vatives of Log and Exponential Functions
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Example: Simplify
a. Ln(      ) =3D        = ;  
e2  
Öe 
e2  
 e1/2<= span style=3D'font-size:178%;color:#006600'>&n= bsp;
Ln(      ) =3D Ln(= e3/2) =3D 3/2
b. e2Ln(x)+1 =3D e2Ln(x) * e =3D (eLn(x))2 * e        = ; 
Deri= vatives of Log and Exponential Functions
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Example: Simplify
a. Ln(      ) =3D        = ;  
e2  
Öe 
e2  
 e1/2<= span style=3D'font-size:178%;color:#006600'>&n= bsp;
Ln(      ) =3D Ln(= e3/2) =3D 3/2
b. e2Ln(x)+1 =3D e2Ln(x) * e =3D (eLn(x))2 * e =3D ex2        
Deri= vatives of Log and Exponential Functions
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Example: Simplify
a. Ln(      ) =3D        = ;  
e2  
Öe 
e2  
 e1/2<= span style=3D'font-size:178%;color:#006600'>&n= bsp;
Ln(      ) =3D Ln(= e3/2) =3D 3/2
b. e2Ln(x)+1 =3D e2Ln(x) * e =3D (eLn(x))2 * e =3D ex2        
Their graphs:
x
y
y =3D ex
Deri= vatives of Log and Exponential Functions
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Example: Simplify
a. Ln(      ) =3D        = ;  
e2  
Öe 
e2  
 e1/2<= span style=3D'font-size:178%;color:#006600'>&n= bsp;
Ln(      ) =3D Ln(= e3/2) =3D 3/2
b. e2Ln(x)+1 =3D e2Ln(x) * e =3D (eLn(x))2 * e =3D ex2        
Their graphs:
x
y
x
y
y =3D ex
y =3D Ln(x)
Deri= vatives of Log and Exponential Functions
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Deri= vatives of Log and Exponential Functions
Uniqness of log and exp function= s:
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Deri= vatives of Log and Exponential Functions
Uniqness of log and exp function= s:
logb(x) =3D logb= (y) x= =3D y  
bx =3D by       x =3D y  
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1. logb(1) =3D 0
2. logb= (x·y) =3D logb(x)+log= b= (y)
3. logb(    ) =3D logb= (x) – logb(y) = ;
4. logb(xt= ) =3D t·logb(x)
x
y
Deri= vatives of Log and Exponential Functions
Uniqness of log and exp function= s:
logb(x) =3D logb= (y) x= =3D y  
bx =3D by       x =3D y  
Rules of log:
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1. logb(1) =3D 0
2. logb= (x·y) =3D logb(x)+log= b= (y)
3. logb(    ) =3D logb= (x) – logb(y) = ;
4. logb(xt= ) =3D t·logb(x)
x
y
(b0 =3D 1)
(br=  · bt =3D br+t)
(      =3D br-t)
((br= )t =3D brt)
bt
br
Deri= vatives of Log and Exponential Functions
Uniqness of log and exp function= s:
logb(x) =3D logb= (y) x= =3D y  
bx =3D by       x =3D y  
Rules of log:
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Deri= vatives of Log and Exponential Functions
Example: Break down the log-expression.= 3;
 
Ln(        = ;          )
3x2 – 5
sin(x) ex
------=_NextPart_01C97CE4.DD467620 Content-Location: file:///C:/AEC9468E/9DerivativeofLogandExponentialFunctions._files/slide0009_image046.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhXAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAABZ AAIAgAAAAAAAAAIMjI+py+0Po5y02lUAADs= ------=_NextPart_01C97CE4.DD467620 Content-Location: file:///C:/AEC9468E/9DerivativeofLogandExponentialFunctions._files/slide0009_image047.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhEQA0AHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAP ADEAgAAAAAAAAAJLhI8XkW3rGoxo0sfu04pvn2nWNVJldDqpFJKt6QEvDIKzerP2zq2JX4nlJEKe aFhBKpQYo6sWVCVvr8XsOkpZl6iJbwu0AmXbY6sAADs= ------=_NextPart_01C97CE4.DD467620 Content-Location: file:///C:/AEC9468E/9DerivativeofLogandExponentialFunctions._files/slide0009_image048.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhYAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAABe AAIAgAAAAAAAAAIMjI+py+0Po5y02loAADs= ------=_NextPart_01C97CE4.DD467620 Content-Location: file:///C:/AEC9468E/9DerivativeofLogandExponentialFunctions._files/slide0038.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Deri= vatives of Log and Exponential Functions
Example: Break down the log-expression.= 3;
 
Ln(        = ;          )
3x2 – 5
sin(x) ex
=3D Ln[(        = ;         )1/2 ]
3x2 – 5
sin(x) ex
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Deri= vatives of Log and Exponential Functions
Example: Break down the log-expression.= 3;
 
Ln(        = ;          )
3x2 – 5
sin(x) ex
=3D Ln[(        = ;         )1/2 ]
3x2 – 5
sin(x) ex
=3D ½ Ln[(<= span style=3D'mso-spacerun:yes'>        = ;         )]
3x2 – 5
sin(x) ex
------=_NextPart_01C97CE4.DD467620 Content-Location: file:///C:/AEC9468E/9DerivativeofLogandExponentialFunctions._files/slide0039_image053.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhXAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAABZ AAIAgAAAAAAAAAIMjI+py+0Po5y02lUAADs= ------=_NextPart_01C97CE4.DD467620 Content-Location: file:///C:/AEC9468E/9DerivativeofLogandExponentialFunctions._files/slide0039_image054.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhEQA0AHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAP ADEAgAAAAAAAAAJLhI8XkW3rGoxo0sfu04pvn2nWNVJldDqpFJKt6QEvDIKzerP2zq2JX4nlJEKe aFhBKpQYo6sWVCVvr8XsOkpZl6iJbwu0AmXbY6sAADs= ------=_NextPart_01C97CE4.DD467620 Content-Location: file:///C:/AEC9468E/9DerivativeofLogandExponentialFunctions._files/slide0039_image055.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhYAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAABe AAIAgAAAAAAAAAIMjI+py+0Po5y02loAADs= ------=_NextPart_01C97CE4.DD467620 Content-Location: file:///C:/AEC9468E/9DerivativeofLogandExponentialFunctions._files/slide0039_image056.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhZgADAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAABk AAIAgAAAAAAAAAIRjI+pyx0Ao5y02nud3rz73xQAOx== ------=_NextPart_01C97CE4.DD467620 Content-Location: file:///C:/AEC9468E/9DerivativeofLogandExponentialFunctions._files/slide0039_image057.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhZgAEAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQBj AAIAgAAAAAAAAAIQjI+py+2vgJy0Wguz3rzDAgA7 ------=_NextPart_01C97CE4.DD467620 Content-Location: file:///C:/AEC9468E/9DerivativeofLogandExponentialFunctions._files/slide0040.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Deri= vatives of Log and Exponential Functions
Example: Break down the log-expression.= 3;
 
Ln(        = ;          )
3x2 – 5
sin(x) ex
=3D Ln[(        = ;         )1/2 ]
3x2 – 5
sin(x) ex
=3D ½ Ln[(<= span style=3D'mso-spacerun:yes'>        = ;         )]
3x2 – 5
sin(x) ex
=3D ½ [Ln(3= x2 – = 5) – Ln(sin(x)ex) ]
------=_NextPart_01C97CE4.DD467620 Content-Location: file:///C:/AEC9468E/9DerivativeofLogandExponentialFunctions._files/slide0040_image058.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhXAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAABZ AAIAgAAAAAAAAAIMjI+py+0Po5y02lUAADs= ------=_NextPart_01C97CE4.DD467620 Content-Location: file:///C:/AEC9468E/9DerivativeofLogandExponentialFunctions._files/slide0040_image059.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhEQA0AHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAP ADEAgAAAAAAAAAJLhI8XkW3rGoxo0sfu04pvn2nWNVJldDqpFJKt6QEvDIKzerP2zq2JX4nlJEKe aFhBKpQYo6sWVCVvr8XsOkpZl6iJbwu0AmXbY6sAADs= ------=_NextPart_01C97CE4.DD467620 Content-Location: file:///C:/AEC9468E/9DerivativeofLogandExponentialFunctions._files/slide0040_image060.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhYAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAABe AAIAgAAAAAAAAAIMjI+py+0Po5y02loAADs= ------=_NextPart_01C97CE4.DD467620 Content-Location: file:///C:/AEC9468E/9DerivativeofLogandExponentialFunctions._files/slide0040_image061.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhZgADAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAABk AAIAgAAAAAAAAAIRjI+pyx0Ao5y02nud3rz73xQAOx== ------=_NextPart_01C97CE4.DD467620 Content-Location: file:///C:/AEC9468E/9DerivativeofLogandExponentialFunctions._files/slide0040_image062.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhZgAEAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQBj AAIAgAAAAAAAAAIQjI+py+2vgJy0Wguz3rzDAgA7 ------=_NextPart_01C97CE4.DD467620 Content-Location: file:///C:/AEC9468E/9DerivativeofLogandExponentialFunctions._files/slide0041.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Deri= vatives of Log and Exponential Functions
Example: Break down the log-expression.= 3;
 
Ln(        = ;          )
3x2 – 5
sin(x) ex
=3D Ln[(        = ;         )1/2 ]
3x2 – 5
sin(x) ex
=3D ½ Ln[(<= span style=3D'mso-spacerun:yes'>        = ;         )]
3x2 – 5
sin(x) ex
=3D ½ [Ln(3= x2 – = 5) – Ln(sin(x)ex) ]
=3D ½ [Ln(3= x2 – = 5) – Ln(sin(x)) – Ln(ex) ]
------=_NextPart_01C97CE4.DD467620 Content-Location: file:///C:/AEC9468E/9DerivativeofLogandExponentialFunctions._files/slide0041_image063.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhXAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAABZ AAIAgAAAAAAAAAIMjI+py+0Po5y02lUAADs= ------=_NextPart_01C97CE4.DD467620 Content-Location: file:///C:/AEC9468E/9DerivativeofLogandExponentialFunctions._files/slide0041_image064.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhEQA0AHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAP ADEAgAAAAAAAAAJLhI8XkW3rGoxo0sfu04pvn2nWNVJldDqpFJKt6QEvDIKzerP2zq2JX4nlJEKe aFhBKpQYo6sWVCVvr8XsOkpZl6iJbwu0AmXbY6sAADs= ------=_NextPart_01C97CE4.DD467620 Content-Location: file:///C:/AEC9468E/9DerivativeofLogandExponentialFunctions._files/slide0041_image065.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhYAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAABe AAIAgAAAAAAAAAIMjI+py+0Po5y02loAADs= ------=_NextPart_01C97CE4.DD467620 Content-Location: file:///C:/AEC9468E/9DerivativeofLogandExponentialFunctions._files/slide0041_image066.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhZgADAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAABk AAIAgAAAAAAAAAIRjI+pyx0Ao5y02nud3rz73xQAOx== ------=_NextPart_01C97CE4.DD467620 Content-Location: file:///C:/AEC9468E/9DerivativeofLogandExponentialFunctions._files/slide0041_image067.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhZgAEAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQBj AAIAgAAAAAAAAAIQjI+py+2vgJy0Wguz3rzDAgA7 ------=_NextPart_01C97CE4.DD467620 Content-Location: file:///C:/AEC9468E/9DerivativeofLogandExponentialFunctions._files/slide0042.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Deri= vatives of Log and Exponential Functions
Example: Break down the log-expression.= 3;
 
Ln(        = ;          )
3x2 – 5
sin(x) ex
=3D Ln[(        = ;         )1/2 ]
3x2 – 5
sin(x) ex
=3D ½ Ln[(<= span style=3D'mso-spacerun:yes'>        = ;         )]
3x2 – 5
sin(x) ex
=3D ½ [Ln(3= x2 – = 5) – Ln(sin(x)ex) ]
=3D ½ [Ln(3= x2 – = 5) – Ln(sin(x)) – Ln(ex) ]
=3D ½ [Ln(3= x2 – = 5) – Ln(sin(x)) – x ]
------=_NextPart_01C97CE4.DD467620 Content-Location: file:///C:/AEC9468E/9DerivativeofLogandExponentialFunctions._files/slide0042_image068.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhXAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAABZ AAIAgAAAAAAAAAIMjI+py+0Po5y02lUAADs= ------=_NextPart_01C97CE4.DD467620 Content-Location: file:///C:/AEC9468E/9DerivativeofLogandExponentialFunctions._files/slide0042_image069.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhEQA0AHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAP ADEAgAAAAAAAAAJLhI8XkW3rGoxo0sfu04pvn2nWNVJldDqpFJKt6QEvDIKzerP2zq2JX4nlJEKe aFhBKpQYo6sWVCVvr8XsOkpZl6iJbwu0AmXbY6sAADs= ------=_NextPart_01C97CE4.DD467620 Content-Location: file:///C:/AEC9468E/9DerivativeofLogandExponentialFunctions._files/slide0042_image070.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhYAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAABe AAIAgAAAAAAAAAIMjI+py+0Po5y02loAADs= ------=_NextPart_01C97CE4.DD467620 Content-Location: file:///C:/AEC9468E/9DerivativeofLogandExponentialFunctions._files/slide0042_image071.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhZgADAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAABk AAIAgAAAAAAAAAIRjI+pyx0Ao5y02nud3rz73xQAOx== ------=_NextPart_01C97CE4.DD467620 Content-Location: file:///C:/AEC9468E/9DerivativeofLogandExponentialFunctions._files/slide0042_image072.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhZgAEAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQBj AAIAgAAAAAAAAAIQjI+py+2vgJy0Wguz3rzDAgA7 ------=_NextPart_01C97CE4.DD467620 Content-Location: file:///C:/AEC9468E/9DerivativeofLogandExponentialFunctions._files/slide0008.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Deri= vatives of Log and Exponential Functions
Example: Solve the following equations.
a. 3e2 – 3x – 2.3 =3D 23.5
------=_NextPart_01C97CE4.DD467620 Content-Location: file:///C:/AEC9468E/9DerivativeofLogandExponentialFunctions._files/slide0043.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Deri= vatives of Log and Exponential Functions
Example: Solve the following equations.
a. 3e2 – 3x – 2.3 =3D 23.5
3e2 – 3x =3D 25.8
------=_NextPart_01C97CE4.DD467620 Content-Location: file:///C:/AEC9468E/9DerivativeofLogandExponentialFunctions._files/slide0044.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Deri= vatives of Log and Exponential Functions
Example: Solve the following equations.
a. 3e2 – 3x – 2.3 =3D 23.5
3e2 – 3x =3D 25.8
e2 – 3x =3D 8.6
------=_NextPart_01C97CE4.DD467620 Content-Location: file:///C:/AEC9468E/9DerivativeofLogandExponentialFunctions._files/slide0045.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Deri= vatives of Log and Exponential Functions
Example: Solve the following equations.
a. 3e2 – 3x – 2.3 =3D 23.5
3e2 – 3x =3D 25.8
e2 – 3x =3D 8.6
2 – 3x =3D L= n 8.6
------=_NextPart_01C97CE4.DD467620 Content-Location: file:///C:/AEC9468E/9DerivativeofLogandExponentialFunctions._files/slide0046.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Deri= vatives of Log and Exponential Functions
Example: Solve the following equations.
a. 3e2 – 3x – 2.3 =3D 23.5
3e2 – 3x =3D 25.8
e2 – 3x =3D 8.6
2 – 3x =3D L= n 8.6
2 – Ln 8.6 = =3D 3x
------=_NextPart_01C97CE4.DD467620 Content-Location: file:///C:/AEC9468E/9DerivativeofLogandExponentialFunctions._files/slide0047.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Deri= vatives of Log and Exponential Functions
Example: Solve the following equations.
a. 3e2 – 3x – 2.3 =3D 23.5
3e2 – 3x =3D 25.8
e2 – 3x =3D 8.6
2 – 3x =3D L= n 8.6
2 – Ln 8.6 = =3D 3x
=3D x
2 – Ln 8.6
3
» x
-0.0506
------=_NextPart_01C97CE4.DD467620 Content-Location: file:///C:/AEC9468E/9DerivativeofLogandExponentialFunctions._files/slide0047_image073.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhbQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAABr AAEAgAAAAAAAAAIJjI+py+0PozQFADs= ------=_NextPart_01C97CE4.DD467620 Content-Location: file:///C:/AEC9468E/9DerivativeofLogandExponentialFunctions._files/slide0048.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Deri= vatives of Log and Exponential Functions
Example: Solve the following equations.
a. 3e2 – 3x – 2.3 =3D 23.5
3e2 – 3x =3D 25.8
e2 – 3x =3D 8.6
2 – 3x =3D L= n 8.6
2 – Ln 8.6 = =3D 3x
=3D x