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text-align: left"><span style='font-size:14.0pt;mso-bidi-font-size:10.0pt; font-family:[SO_GB2312;mso-ascii-font-family:"Times New Roman"'> N0cpeQpeN[peQpe</span></span> <span style='font-size:14.0pt;mso-bidi-font-size:10.0pt; font-family:[SO_GB2312;mso-ascii-font-family:"Times New Roman"'>Third, the exponential function and logarithmic functions</span></span> <span lang=EN-US style=font-size:14.0pt;mso-bidi-font-size:10.0pt></span></p><p class=MsoNormal> <span class="notranslate" onmouseover="_tipon(this)" onmouseout="_tipoff()"><span class="google-src-text" style="direction: ltr; text-align: left"><span lang=EN-US>[</span> <span style='font-family:[SO_GB2312;mso-ascii-font-family:"Times New Roman"'>[IN</span> <span lang=EN-US>]</span></span> <span style='font-family:[SO_GB2312;mso-ascii-font-family:"Times New Roman"'><span lang=EN-US>[Definition]</span></span></span> <span lang=EN-US><span style="mso-spacerun: yes">&nbsp; </span></span> <span class="notranslate" onmouseover="_tipon(this)" onmouseout="_tipoff()"><span class="google-src-text" style="direction: ltr; text-align: left"><span style='font-family:[SO_GB2312;mso-ascii-font-family:"Times New Roman"'>b_Y</span></span> <span style='font-family:[SO_GB2312;mso-ascii-font-family:"Times New Roman"'>Shaped like</span></span> <sub><span lang=EN-US><img width=217 height=24 src=./1.files/image002.gif v:shapes=_x0000_i1025 align=absmiddle></span></sub> <span class="notranslate" onmouseover="_tipon(this)" onmouseout="_tipoff()"><span class="google-src-text" style="direction: ltr; text-align: left"><span style='font-family:[SO_GB2312;mso-ascii-font-family: "Times New Roman"'>vQpey:NcpeQpe</span> <span lang=EN-US>.</span></span> <span style='font-family:[SO_GB2312;mso-ascii-font-family: "Times New Roman"'>The function is called exponential <span lang=EN-US>function.</span></span></span> </p><p class=MsoNormal><span class="notranslate" onmouseover="_tipon(this)" onmouseout="_tipoff()"><span class="google-src-text" style="direction: ltr; text-align: left"><span style='font-family:[SO_GB2312; mso-ascii-font-family:"Times New Roman"'>S_</span> <span lang=EN-US><i style="mso-bidi-font-style: normal">a</i> = <i style="mso-bidi-font-style: normal">e</i></span> <span style='font-family:[SO_GB2312;mso-ascii-font-family: "Times New Roman"'>e :NfNQeO geb</span></span> <span style='font-family:[SO_GB2312; mso-ascii-font-family:"Times New Roman"'>When</span> <span lang=EN-US><i style="mso-bidi-font-style: normal">a</i> =</span> <span style='font-family:[SO_GB2312;mso-ascii-font-family: "Times New Roman"'><span lang=EN-US><i style="mso-bidi-font-style: normal">e,</i></span> in order to facilitate the writing, sometimes</span></span> <sub><span lang=EN-US><img width=18 height=20 src=./1.files/image004.gif v:shapes=_x0000_i1026 align=absmiddle></span></sub> <span class="notranslate" onmouseover="_tipon(this)" onmouseout="_tipoff()"><span class="google-src-text" style="direction: ltr; text-align: left"><span style='font-family:[SO_GB2312;mso-ascii-font-family: "Times New Roman"'>\O</span> <span lang=EN-US>exp <i style="mso-bidi-font-style: normal">x</i></span> <span style='font-family:[SO_GB2312;mso-ascii-font-family: "Times New Roman"'> b</span></span> <span style='font-family:[SO_GB2312;mso-ascii-font-family: "Times New Roman"'>Denoted by</span> <span lang=EN-US>exp</span> <span style='font-family:[SO_GB2312;mso-ascii-font-family: "Times New Roman"'><span lang=EN-US><i style="mso-bidi-font-style: normal">x,</i></span> the</span></span> <sub><span lang=EN-US><img width=33 height=20 src=./1.files/image006.gif v:shapes=_x0000_i1027 align=absmiddle></span></sub> <span class="notranslate" onmouseover="_tipon(this)" onmouseout="_tipoff()"><span class="google-src-text" style="direction: ltr; text-align: left"><span style='font-family:[SO_GB2312;mso-ascii-font-family: "Times New Roman"'>\O</span> <span lang=EN-US>exp{ <i style="mso-bidi-font-style: normal">f</i> ( <i style=mso-bidi-font-style:normal>x</i> )}</span> <span style='font-family:[SO_GB2312;mso-ascii-font-family:"Times New Roman"'> I{I{</span> <span lang=EN-US>.</span></span> <span style='font-family:[SO_GB2312;mso-ascii-font-family: "Times New Roman"'>Denoted by</span> <span lang=EN-US>exp <i style="mso-bidi-font-style: normal">{f</i></span> <span style='font-family:[SO_GB2312;mso-ascii-font-family:"Times New Roman"'><span lang=EN-US><i style=mso-bidi-font-style:normal>(x)},</i></span> and the <span lang=EN-US>like.</span></span></span> </p><p class=MsoNormal> <span class="notranslate" onmouseover="_tipon(this)" onmouseout="_tipoff()"><span class="google-src-text" style="direction: ltr; text-align: left"><span style='font-family:[SO_GB2312; mso-ascii-font-family:"Times New Roman"'>(WQpesQ|_</span></span> <span style='font-family:[SO_GB2312; mso-ascii-font-family:"Times New Roman"'>In the function formula</span></span> <sub><span lang=EN-US><img width=44 height=21 src=./1.files/image008.gif v:shapes=_x0000_i1028 align=absmiddle></span></sub> <span class="notranslate" onmouseover="_tipon(this)" onmouseout="_tipoff()"><span class="google-src-text" style="direction: ltr; text-align: left"><span style='font-family:[SO_GB2312;mso-ascii-font-family: "Times New Roman"'>-N</span></span> <span style='font-family:[SO_GB2312;mso-ascii-font-family: "Times New Roman"'>In</span></span> <sub><span lang=EN-US><img width=148 height=21 src=./1.files/image010.gif v:shapes=_x0000_i1029 align=absmiddle></span></sub> <span class="notranslate" onmouseover="_tipon(this)" onmouseout="_tipoff()"><span class="google-src-text" style="direction: ltr; text-align: left"><span style='font-family:[SO_GB2312;mso-ascii-font-family: "Times New Roman"'> 傊b</span> <i style=mso-bidi-font-style:normal><span lang=EN-US>x</span></i> <span style='font-family:[SO_GB2312;mso-ascii-font-family: "Times New Roman"'>Ɖ:NSϑ </span> <i style=mso-bidi-font-style:normal><span lang=EN-US>y</span></i> <span style='font-family:[SO_GB2312;mso-ascii-font-family: "Times New Roman"'>Ɖ:NVSϑ Ry</span> <i style=mso-bidi-font-style:normal><span lang=EN-US>y</span></i> <span style='font-family:[SO_GB2312;mso-ascii-font-family: "Times New Roman"'>/fN</span> <i style=mso-bidi-font-style:normal><span lang=EN-US>a</span></i> <span style='font-family:[SO_GB2312;mso-ascii-font-family: "Times New Roman"'>:N^v</span> <i style=mso-bidi-font-style:normal><span lang=EN-US>x</span></i> <span style='font-family:[SO_GB2312;mso-ascii-font-family: "Times New Roman"'>v[peQpe </span> <i style=mso-bidi-font-style:normal><span lang=EN-US>x</span></i> <span style='font-family:[SO_GB2312;mso-ascii-font-family: "Times New Roman"'>y:Nwpe \O</span></span> <span style='font-family:[SO_GB2312;mso-ascii-font-family: "Times New Roman"'>If treated as the independent variable <i style=mso-bidi-font-style:normal><span lang=EN-US>x,</span></i></span> <i style=mso-bidi-font-style:normal><span lang=EN-US>y</span></i> <span style='font-family:[SO_GB2312;mso-ascii-font-family: "Times New Roman"'>regarded as the dependent variable,</span> <i style=mso-bidi-font-style:normal><span lang=EN-US>y</span></i> <span style='font-family:[SO_GB2312;mso-ascii-font-family: "Times New Roman"'>is called as</span> <i style=mso-bidi-font-style:normal><span lang=EN-US>a</span></i> <span style='font-family:[SO_GB2312;mso-ascii-font-family: "Times New Roman"'>logarithm function of <i style=mso-bidi-font-style:normal><span lang=EN-US>x,</span></i></span> <i style=mso-bidi-font-style:normal><span lang=EN-US>x</span></i> <span style='font-family:[SO_GB2312;mso-ascii-font-family: "Times New Roman"'>is called the real number, referred to as</span></span> <sub><span lang=EN-US><img width=71 height=21 src=./1.files/image012.gif v:shapes=_x0000_i1030 align=absmiddle></span></sub> <span class="notranslate" onmouseover="_tipon(this)" onmouseout="_tipoff()"><span class="google-src-text" style="direction: ltr; text-align: left"><span lang=EN-US>.</span> <span style='font-family:[SO_GB2312;mso-ascii-font-family: "Times New Roman"'>cpeQpeT[peQpeN:NSQpe</span> <span lang=EN-US>.</span></span> <span lang=EN-US>.</span> <span style='font-family:[SO_GB2312;mso-ascii-font-family: "Times New Roman"'>Exponential and logarithmic functions mutually inverse <span lang=EN-US>functions.</span></span></span> </p><p class=MsoNormal> <span class="notranslate" onmouseover="_tipon(this)" onmouseout="_tipoff()"><span class="google-src-text" style="direction: ltr; text-align: left"><span lang=EN-US>[</span> <span style='font-family:[SO_GB2312;mso-ascii-font-family:"Times New Roman"'>QpeVb_Nyr_</span> <span lang=EN-US>]</span></span> <span style='font-family:[SO_GB2312;mso-ascii-font-family:"Times New Roman"'><span lang=EN-US>[Function</span> graphics and <span lang=EN-US>features]</span></span></span> </p><table border=1 cellspacing=0 cellpadding=0 style=" border-collapse:collapse;border:none;mso-border-alt:solid windowtext .5pt; mso-padding-alt:0cm 5.4pt 0cm 5.4pt"><tr><td width=227 valign=top style="width:6.0cm;border:solid windowtext .5pt; border-left:none;padding:0cm 5.4pt 0cm 5.4pt"><p class=MsoNormal align=center style=text-align:center> <span class="notranslate" onmouseover="_tipon(this)" onmouseout="_tipoff()"><span class="google-src-text" style="direction: ltr; text-align: left"><span style='font-family:[SO_GB2312;mso-ascii-font-family:"Times New Roman"'>e zNVb_</span></span> <span style='font-family:[SO_GB2312;mso-ascii-font-family:"Times New Roman"'>Equations and graphics</span></span> </p></td> <td width=416 valign=top style="width:11.0cm;border-top:solid windowtext .5pt; border-left:none;border-bottom:solid windowtext .5pt;border-right:none; mso-border-left-alt:solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt"><p class=MsoNormal align=center style=text-align:center> <span class="notranslate" onmouseover="_tipon(this)" onmouseout="_tipoff()"><span class="google-src-text" style="direction: ltr; text-align: left"><span style='font-family:[SO_GB2312;mso-ascii-font-family:"Times New Roman"'>yr</span></span> <span style='font-family:[SO_GB2312;mso-ascii-font-family:"Times New Roman"'>Special</span></span> <span lang=EN-US><span style="mso-spacerun: yes">&nbsp;&nbsp; </span></span> <span class="notranslate" onmouseover="_tipon(this)" onmouseout="_tipoff()"><span class="google-src-text" style="direction: ltr; text-align: left"><span style='font-family:[SO_GB2312;mso-ascii-font-family:"Times New Roman"'>_</span></span> <span style='font-family:[SO_GB2312;mso-ascii-font-family:"Times New Roman"'>Levy</span></span> </p></td></tr><tr><td width=227 valign=top style="width:6.0cm;border-top:none;border-left:none; border-bottom:solid windowtext .5pt;border-right:solid windowtext .5pt; mso-border-top-alt:solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt"><p class=MsoNormal> <span class="notranslate" onmouseover="_tipon(this)" onmouseout="_tipoff()"><span class="google-src-text" style="direction: ltr; text-align: left"><span style='font-family:[SO_GB2312;mso-ascii-font-family: "Times New Roman"'>cpeQpe</span></span> <span style='font-family:[SO_GB2312;mso-ascii-font-family: "Times New Roman"'>Exponential function</span></span> <span lang=EN-US></span></p><p class=MsoNormal> <span lang=EN-US><span style="mso-spacerun: yes">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span> <sub><img width=92 height=24 src=./1.files/image014.gif v:shapes=_x0000_i1075></sub></span> </p><p class=MsoNormal align=center style=text-align:center> <span lang=EN-US><img width=222 height=178 src=./1.files/image016.jpg v:shapes=_x0000_i1040></span> </p></td><td width=416 style="width:11.0cm;border:none;border-bottom:solid windowtext .5pt; mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid windowtext .5pt; padding:0cm 5.4pt 0cm 5.4pt"><p class=MsoNormal> <span class="notranslate" onmouseover="_tipon(this)" onmouseout="_tipoff()"><span class="google-src-text" style="direction: ltr; text-align: left"><span style='font-family:[SO_GB2312; mso-ascii-font-family:"Times New Roman"'>f~N</span> <i style="mso-bidi-font-style: normal"><span lang=EN-US>y</span></i> <span style='font-family:[SO_GB2312; mso-ascii-font-family:"Times New Roman"'>tvNNp</span> <span lang=EN-US><i style="mso-bidi-font-style: normal">A</i> (0,1).</span></span> <span style='font-family:[SO_GB2312; mso-ascii-font-family:"Times New Roman"'>Curve and</span> <i style="mso-bidi-font-style: normal"><span lang=EN-US>the</span></i> <span style='font-family:[SO_GB2312; mso-ascii-font-family:"Times New Roman"'><i style="mso-bidi-font-style: normal"><span lang=EN-US>y-axis</span></i> at point</span> <span lang=EN-US><i style="mso-bidi-font-style: normal">A</i> (0,1).</span></span> </p><p class=MsoNormal> <span class="notranslate" onmouseover="_tipon(this)" onmouseout="_tipoff()"><span class="google-src-text" style="direction: ltr; text-align: left"><span style='font-family:[SO_GB2312; mso-ascii-font-family:"Times New Roman"'>nя~:N</span> <span lang=EN-US><i style="mso-bidi-font-style: normal">y</i> =0.</span></span> <span style='font-family:[SO_GB2312; mso-ascii-font-family:"Times New Roman"'>Asymptote of</span> <span lang=EN-US><i style="mso-bidi-font-style: normal">y</i> = 0.</span></span> </p></td></tr><tr><td width=227 valign=top style="width:6.0cm;border-top:none;border-left:none; border-bottom:solid windowtext .5pt;border-right:solid windowtext .5pt; mso-border-top-alt:solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt"><p class=MsoNormal> <span class="notranslate" onmouseover="_tipon(this)" onmouseout="_tipoff()"><span class="google-src-text" style="direction: ltr; text-align: left"><span style='font-family:[SO_GB2312;mso-ascii-font-family: "Times New Roman"'>[peQpe</span></span> <span style='font-family:[SO_GB2312;mso-ascii-font-family: "Times New Roman"'>Logarithmic function</span></span> </p><p class=MsoNormal> <span lang=EN-US><span style="mso-spacerun: yes">&nbsp; </span> <span style=mso-text-raise:-6.0pt><sub><img width=111 height=27 src=./1.files/image018.gif v:shapes=_x0000_i1109></sub></span></span> </p><p class=MsoNormal> <span lang=EN-US><img width=222 height=184 src=./1.files/image020.jpg v:shapes=_x0000_i1069></span> </p></td><td width=416 style="width:11.0cm;border:none;border-bottom:solid windowtext .5pt; mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid windowtext .5pt; padding:0cm 5.4pt 0cm 5.4pt"><p class=MsoNormal> <span class="notranslate" onmouseover="_tipon(this)" onmouseout="_tipoff()"><span class="google-src-text" style="direction: ltr; text-align: left"><span style='font-family:[SO_GB2312; mso-ascii-font-family:"Times New Roman"'>f~N</span> <i style="mso-bidi-font-style: normal"><span lang=EN-US>x</span></i> <span style='font-family:[SO_GB2312; mso-ascii-font-family:"Times New Roman"'>tvNNp</span> <span lang=EN-US><i style="mso-bidi-font-style: normal">A</i> (1,0).</span></span> <span style='font-family:[SO_GB2312; mso-ascii-font-family:"Times New Roman"'>Curve intersects with</span> <i style="mso-bidi-font-style: normal"><span lang=EN-US>the x</span></i> <span style='font-family:[SO_GB2312; mso-ascii-font-family:"Times New Roman"'>axis at point</span> <span lang=EN-US><i style="mso-bidi-font-style: normal">A</i> (1,0).</span></span> </p><p class=MsoNormal> <span class="notranslate" onmouseover="_tipon(this)" onmouseout="_tipoff()"><span class="google-src-text" style="direction: ltr; text-align: left"><span style='font-family:[SO_GB2312; mso-ascii-font-family:"Times New Roman"'>nя~:N</span> <span lang=EN-US><i style="mso-bidi-font-style: normal">x</i> =0.</span></span> <span style='font-family:[SO_GB2312; mso-ascii-font-family:"Times New Roman"'>Asymptote for</span> <span lang=EN-US><i style="mso-bidi-font-style: normal">x</i> = 0.</span></span> </p></td></tr></table><p class=MsoNormal> <span class="notranslate" onmouseover="_tipon(this)" onmouseout="_tipoff()"><span class="google-src-text" style="direction: ltr; text-align: left"><span lang=EN-US>[</span> <span style='font-family:[SO_GB2312;mso-ascii-font-family:"Times New Roman"'>cpeЏ{lR</span> <span lang=EN-US>]</span></span> <span lang=EN-US><span style='font-family:[SO_GB2312;mso-ascii-font-family:"Times New Roman"'>[Index algorithm]</span></span></span> </p><p class=MsoNormal align=center style=text-align:center> <sub><span lang=EN-US><img width=267 height=147 src=./1.files/image022.gif v:shapes=_x0000_i1034></span></sub> </p><p class=MsoNormal> <span class="notranslate" onmouseover="_tipon(this)" onmouseout="_tipoff()"><span class="google-src-text" style="direction: ltr; text-align: left"><span lang=EN-US>[</span> <span style='font-family:[SO_GB2312;mso-ascii-font-family:"Times New Roman"'>[pev'`(NЏ{lR</span> <span lang=EN-US>]</span></span> <span style='font-family:[SO_GB2312;mso-ascii-font-family:"Times New Roman"'><span lang=EN-US>[Nature</span> and algorithms <span lang=EN-US>logarithm]</span></span></span> <span lang=EN-US><span style="mso-spacerun: yes">&nbsp; </span></span> <span class="notranslate" onmouseover="_tipon(this)" onmouseout="_tipoff()"><span class="google-src-text" style="direction: ltr; text-align: left"><span style='font-family:[SO_GB2312;mso-ascii-font-family:"Times New Roman"'>(W NbvlQ_-N GP</span> <span lang=EN-US><i style=mso-bidi-font-style:normal>a</i> &gt;0</span> <span style='font-family:[SO_GB2312;mso-ascii-font-family: "Times New Roman"'> Te@bG0RvQpeGP/f(W[INW̑v</span> <span lang=EN-US>.</span></span> <span style='font-family:[SO_GB2312;mso-ascii-font-family:"Times New Roman"'>In the following formula, assuming</span> <span lang=EN-US><i style=mso-bidi-font-style:normal>a&gt;</i></span> <span style='font-family:[SO_GB2312;mso-ascii-font-family: "Times New Roman"'><span lang=EN-US>0,</span> while the function are assumed to be encountered in the definition domain of <span lang=EN-US>discussion.</span></span></span> </p><p class=MsoNormal> <span lang=EN-US><span style=mso-tab-count:1>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span> <span class="notranslate" onmouseover="_tipon(this)" onmouseout="_tipoff()"><span class="google-src-text" style="direction: ltr; text-align: left"><span style='font-family:[SO_GB2312;mso-ascii-font-family:"Times New Roman"'>Npel g[pe</span></span> <span style='font-family:[SO_GB2312;mso-ascii-font-family:"Times New Roman"'>No zero and negative logarithm</span></span> <span lang=EN-US><span style=mso-tab-count:1>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span> <sub><img width=67 height=21 src=./1.files/image024.gif v:shapes=_x0000_i1047 align=absmiddle></sub></span> </p><p class=MsoNormal> <span lang=EN-US><span style=mso-tab-count:1>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span> <sub><img width=65 height=21 src=./1.files/image026.gif v:shapes=_x0000_i1048></sub> <span style=mso-tab-count:1>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span> <sub><img width=163 height=21 src=./1.files/image028.gif v:shapes=_x0000_i1049></sub></span> </p><p class=MsoNormal> <span lang=EN-US><span style=mso-tab-count:1>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span> <sub><img width=159 height=44 src=./1.files/image030.gif v:shapes=_x0000_i1050 align=absmiddle></sub> <span style=mso-tab-count:1>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span> <sub><img width=123 height=24 src=./1.files/image032.gif v:shapes=_x0000_i1051 align=absmiddle></sub></span> </p><p class=MsoNormal> <span lang=EN-US><span style=mso-tab-count:1>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span> <span class="notranslate" onmouseover="_tipon(this)" onmouseout="_tipoff()"><span class="google-src-text" style="direction: ltr; text-align: left"><span style='font-family:[SO_GB2312;mso-ascii-font-family:"Times New Roman"'>[peR`I{_</span></span> <span style='font-family:[SO_GB2312;mso-ascii-font-family:"Times New Roman"'>Logarithmic identities</span></span> <span lang=EN-US><sub><img width=67 height=24 src=./1.files/image034.gif v:shapes=_x0000_i1052 align=absmiddle></sub> <span style=mso-tab-count:1>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span> <span class="notranslate" onmouseover="_tipon(this)" onmouseout="_tipoff()"><span class="google-src-text" style="direction: ltr; text-align: left"><span style='font-family:[SO_GB2312;mso-ascii-font-family:"Times New Roman"'>bc^lQ_</span></span> <span style='font-family:[SO_GB2312;mso-ascii-font-family:"Times New Roman"'>Bottom change formula</span></span> <sub><span lang=EN-US><img width=103 height=45 src=./1.files/image036.gif v:shapes=_x0000_i1053 align=absmiddle></span></sub> </p><p class=MsoNormal> <span lang=EN-US><span style=mso-tab-count:1>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span> <sub><img width=112 height=21 src=./1.files/image038.gif v:shapes=_x0000_i1054></sub></span> </p><p class=MsoNormal> <span class="notranslate" onmouseover="_tipon(this)" onmouseout="_tipoff()"><span class="google-src-text" style="direction: ltr; text-align: left"><span lang=EN-US>[</span> <span style='font-family:[SO_GB2312;mso-ascii-font-family: "Times New Roman"'>8^(u[peN6q[pe</span> <span lang=EN-US>]</span></span> <span style='font-family:[SO_GB2312;mso-ascii-font-family: "Times New Roman"'><span lang=EN-US>[Common</span> logarithm of the natural <span lang=EN-US>logarithm]</span></span></span> </p><p class=MsoNormal> <span class="notranslate" onmouseover="_tipon(this)" onmouseout="_tipoff()"><span class="google-src-text" style="direction: ltr; text-align: left"><span lang=EN-US>1 <sup>o</sup></span></span> <span lang=EN-US>1 <sup>o</sup></span></span> <span lang=EN-US><span style="mso-spacerun: yes">&nbsp; </span></span> <span class="notranslate" onmouseover="_tipon(this)" onmouseout="_tipoff()"><span class="google-src-text" style="direction: ltr; text-align: left"><span style='font-family:[SO_GB2312;mso-ascii-font-family:"Times New Roman"'>8^(u[peN</span> <span lang=EN-US>10</span> <span style='font-family:[SO_GB2312;mso-ascii-font-family: "Times New Roman"'>:N^v[pey:N8^(u[pe \O</span></span> <span style='font-family:[SO_GB2312;mso-ascii-font-family:"Times New Roman"'>Common logarithm:</span> <span style='font-family:[SO_GB2312;mso-ascii-font-family: "Times New Roman"'>logarithm</span> <span style='font-family:[SO_GB2312;mso-ascii-font-family:"Times New Roman"'>to</span> <span style='font-family:[SO_GB2312;mso-ascii-font-family: "Times New Roman"'>the base</span> <span lang=EN-US>10</span> <span style='font-family:[SO_GB2312;mso-ascii-font-family: "Times New Roman"'>called the common logarithm, denoted</span></span> </p><p class=MsoNormal> <span lang=EN-US><span style="mso-spacerun: yes">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span><span style="mso-spacerun: yes">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</span> <sub><img width=88 height=21 src=./1.files/image040.gif v:shapes=_x0000_i1055></sub></span> </p><p class=MsoNormal> <span class="notranslate" onmouseover="_tipon(this)" onmouseout="_tipoff()"><span class="google-src-text" style="direction: ltr; text-align: left"><span lang=EN-US>2 <sup>o</sup></span></span> <span lang=EN-US>2 <sup>o</sup></span></span> <span lang=EN-US><span style="mso-spacerun: yes">&nbsp; </span></span> <span class="notranslate" onmouseover="_tipon(this)" onmouseout="_tipoff()"><span class="google-src-text" style="direction: ltr; text-align: left"><span style='font-family:[SO_GB2312;mso-ascii-font-family:"Times New Roman"'>6q[peN</span> <span lang=EN-US><i style=mso-bidi-font-style:normal>e</i> =2.718281828459</span> <span style='mso-char-type: symbol;mso-symbol-font-family:"MT Extra"'><span lang=EN-US style='font-family:"MT Extra"; mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Roman"; mso-char-type:symbol;mso-symbol-font-family:"MT Extra"'>L</span></span> <span style='font-family:[SO_GB2312;mso-ascii-font-family:"Times New Roman"'>:N^v[pey:N6q[pe \O</span></span> <span style='font-family:[SO_GB2312;mso-ascii-font-family:"Times New Roman"'>Natural logarithm: logarithm of</span> <span lang=EN-US><i style=mso-bidi-font-style:normal>e</i> = <i style=mso-bidi-font-style:normal>2.718281828459</i></span> <span style='mso-char-type: symbol;mso-symbol-font-family:"MT Extra"'><span lang=EN-US style='font-family:"MT Extra"; mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Roman"; mso-char-type:symbol;mso-symbol-font-family:"MT Extra"'>L</span></span> <span style='font-family:[SO_GB2312;mso-ascii-font-family:"Times New Roman"'>called for the end of the natural logarithm, denoted</span></span> </p><p class=MsoNormal align=center style="text-align:center; "> <sub><span lang=EN-US><img width=83 height=21 src=./1.files/image042.gif v:shapes=_x0000_i1056></span></sub> </p><p class=MsoNormal> <span class="notranslate" onmouseover="_tipon(this)" onmouseout="_tipoff()"><span class="google-src-text" style="direction: ltr; text-align: left"><span lang=EN-US>3 <sup>o</sup></span></span> <span lang=EN-US>3 <sup>o</sup></span></span> <span lang=EN-US><span style="mso-spacerun: yes">&nbsp; </span></span> <span class="notranslate" onmouseover="_tipon(this)" onmouseout="_tipoff()"><span class="google-src-text" style="direction: ltr; text-align: left"><span style='font-family:[SO_GB2312;mso-ascii-font-family:"Times New Roman"'>8^(u[peN6q[pevsQ|</span></span> <span style='font-family:[SO_GB2312;mso-ascii-font-family:"Times New Roman"'>Common logarithm of the natural logarithm of the relationship:</span></span> </p><p class=MsoNormal align=center style="text-align:center; "> <sub><span lang=EN-US><img width=187 height=41 src=./1.files/image044.gif v:shapes=_x0000_i1070></span></sub> </p><p class=MsoNormal><span class="notranslate" onmouseover="_tipon(this)" onmouseout="_tipoff()"><span class="google-src-text" style="direction: ltr; text-align: left"><span style='font-family: [SO_GB2312;mso-ascii-font-family:"Times New Roman"'>_-N</span> <span lang=EN-US><i style=mso-bidi-font-style:normal>M</i></span> <span style='font-family:[SO_GB2312;mso-ascii-font-family:"Times New Roman"'>y:N!jpe </span></span> <span style='font-family: [SO_GB2312;mso-ascii-font-family:"Times New Roman"'>Where <span lang=EN-US><i style=mso-bidi-font-style:normal>M</i></span></span> <span style='font-family:[SO_GB2312;mso-ascii-font-family:"Times New Roman"'>is called the modulus,</span></span> </p><p class=MsoNormal align=center style=text-align:center;> <span lang=EN-US><span style="mso-spacerun: yes">&nbsp;&nbsp;&nbsp; </span> <sub><img width=201 height=64 src=./1.files/image046.gif v:shapes=_x0000_i1071></sub></span> </p><p class=MsoNormal> <span class="notranslate" onmouseover="_tipon(this)" onmouseout="_tipoff()"><span class="google-src-text" style="direction: ltr; text-align: left"><span lang=EN-US>4 <sup>o</sup></span></span> <span lang=EN-US>4 <sup>o</sup></span></span> <span lang=EN-US><span style="mso-spacerun: yes">&nbsp; </span></span> <span class="notranslate" onmouseover="_tipon(this)" onmouseout="_tipoff()"><span class="google-src-text" style="direction: ltr; text-align: left"><span style='font-family:[SO_GB2312;mso-ascii-font-family:"Times New Roman"'>8^(u[pepeBll</span></span> <span style='font-family:[SO_GB2312;mso-ascii-font-family:"Times New Roman"'>Common logarithm method for finding the first few:</span></span> </p><p class=MsoNormal> <span class="notranslate" onmouseover="_tipon(this)" onmouseout="_tipoff()"><span class="google-src-text" style="direction: ltr; text-align: left"><span style='font-family:[SO_GB2312;mso-ascii-font-family:"Times New Roman"'>wpe'YN</span> <span lang=EN-US>1</span> <span style='font-family:[SO_GB2312;mso-ascii-font-family: "Times New Roman"'> R[pevpe:Nckpeb vQ <span lang=EN-US>1.</span></span> <span style='font-family:[SO_GB2312;mso-ascii-font-family:"Times New Roman"'>If that number is greater than</span> <span style='font-family:[SO_GB2312;mso-ascii-font-family: "Times New Roman"'><span lang=EN-US>1,</span> then for the first few numbers of positive or zero, its value is less than the integer digits</span> <span lang=EN-US>1.</span></span> </p><p class=MsoNormal> <span class="notranslate" onmouseover="_tipon(this)" onmouseout="_tipoff()"><span class="google-src-text" style="direction: ltr; text-align: left"><span style='font-family:[SO_GB2312;mso-ascii-font-family:"Times New Roman"'>wpe\N</span> <span lang=EN-US>1</span> <span style='font-family:[SO_GB2312;mso-ascii-font-family: "Times New Roman"'> R[pevpe:Npe vQ~[ <span lang=EN-US>0</span> <span style='font-family:[SO_GB2312;mso-ascii-font-family:"Times New Roman"'> v*Npe</span> <span lang=EN-US>(</span> <span style='font-family:[SO_GB2312;mso-ascii-font-family: "Times New Roman"'>Sb\pepMRv*N </span> <span lang=EN-US>0</span> <span style='font-family:[SO_GB2312;mso-ascii-font-family:"Times New Roman"'> </span> <span lang=EN-US>).</span></span> <span style='font-family:[SO_GB2312;mso-ascii-font-family:"Times New Roman"'>If that number is less than</span> <span style='font-family:[SO_GB2312;mso-ascii-font-family: "Times New Roman"'><span lang=EN-US>1,</span> then for the first count is negative, its absolute value is equal to the true number of significant figures in front of the first <span style='font-family:[SO_GB2312;mso-ascii-font-family:"Times New Roman"'><span lang=EN-US>&quot;0&quot;</span></span></span> <span style='font-family:[SO_GB2312;mso-ascii-font-family:"Times New Roman"'>number</span> <span style='font-family:[SO_GB2312;mso-ascii-font-family: "Times New Roman"'><span lang=EN-US>(including</span> the decimal point in front of the <span style='font-family:[SO_GB2312;mso-ascii-font-family:"Times New Roman"'><span lang=EN-US>&quot;0&quot;).</span></span></span></span> </p><p class=MsoNormal> <span class="notranslate" onmouseover="_tipon(this)" onmouseout="_tipoff()"><span class="google-src-text" style="direction: ltr; text-align: left"><span style='font-family:[SO_GB2312;mso-ascii-font-family:"Times New Roman"'>[pev>\pe1u[pehgQ</span> <span lang=EN-US>.</span></span> <span style='font-family:[SO_GB2312;mso-ascii-font-family:"Times New Roman"'>Logarithmic mantissa detected by logarithmic <span lang=EN-US>tables.</span></span></span> </p></div></body> <script>_addload(function(){_setupIW();_csi('zh-CN','en','http://drhuang.com/chinese/science/mathematics/handbook/a23/a23.htm');});</script></html>