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D1E@*88,~Gdeqe"* 8?88]HXa`` TGdeqe"* 8@88^HXn`` TB ` 0C t$ deqe  u$ deqe D1E@!0~,~Gdeqe"* A88_HX a`^0 ``=``1`` VGdeqe"* NB88`HX$a`^{-`n`} ``=`` 1 over {a`^n `} `` ?B `0D1E@:R,~B `0C3. pX 1 D1E@?R,~B `!0C  ( deqe t, deqe t X ) D1E@-D88,~Gdeqe"* 8C88aHX a``>``0``,``b``>``0`` TGdeqe"*  8D88bHX m``,``n`` TB `! 0C  t$ deqe  u$ deqe D1E@I ,~Gdeqe"* E88cHX a^m`a^n`=`a^{m`+`n`}` VGdeqe"* ]F88dHX(a^m`)^n`=`a^{m`n}` VB `0 0C v$ deqe C w$ deqe (, deqe ) D1E@sO6 6 ,~Gdeqe"* ]G88eHX(a`b)^m`=`a^m`b^m` VGdeqe"* 6 H88fHX+left( a over b`right)^m`=`{a^m}over{b^m}` AGdeqe"* 8I88gHX b`neq`0` TB `! 0C  x$ deqe  y$ deqe D1E@Y6 6 ,~Gdeqe"* 6 J88hHX.{a`^m `} over {a`^n `} ``=``a`^{m``-``n`} `` AGdeqe"* zK88iHX$a`^{-`m`} ``=`` 1 over {a`^m `} `` ?O ZZP,P,B`DE@RXB`DE@@RXB`DE@%RXB`DE@+RXB`DE@2RXB`DE@@8RXB`DE@>RXB`DE@DRXB`DE@KRXB`DE@@QRXB`DE@WRXB`DE@]RXB`DE@dRXB`DE@@jRXB`DE@pRXB`DE@vRXB`DE@}RXB`DE@@RXB`DE@RXB` C  osg DE@gXG osgCjqGL` noc$noc$qq 8 ??????noc$cer$cer$L noc$7zq 8 ?K@?????\\?c#{???cip$lle$Lcip$7ȯ z$t= ?K@?????\\?c#{???V_? ` `???Uȯȯ  ȯ Llle$I 8!8! \\?@?@????\\?&gc#{?N[Q??+?+???!@H  B`C1-1 DE@` `  P8! and elsewhere.L cer$@ @ ??t@?????PE???!@H @ B`C` DE@RX O@ @ L cer$XXD,l?@?x@????????!@H XB`DE@RXPVOXXB`CL  3@ @?  ne D E@7RXG ne)HB` C `onta c$ D E @RXG onta)B`C@`$ 64X 8@ deqe пt. DE@w<<XGdeqe"*  <88͢HX `^3root64 VB`C a$ -8X 8  x @ . DE@ RXB` Cb$ 8X $  x @ deqe t. DE@K<<XGdeqe"* <88΢HX `^4 root8 VB`3C`c$ deqe t @| L, 3X deqe   x @ \ t. DE@߿XGdeqe"* 88ϢHXn` TGdeqe"* 88ТHXn` TB`4Cd$ deqe t | L, -4X deqe   x @ P t. DE@XGdeqe"* 88ѢHXn` TGdeqe"*  88ҢHXn` TB`DE@RXB`C00 D#E@/RXB`DE@oRXB`DE@RXB`DE@RXB`DE@/RXB`DE@oRXB`DE@RXB` C@ osg D(E@gXG osgCjqGL` noc$noc$qq 8 ??????noc$cer$cer$L noc$7zq 8 ?K@?????\\?c#{???cip$lle$Lcip$7ȯ z$t= ?K@?????\\?c#{???V_? ` `???Uȯȯ  ȯ Llle$I 8!8! \\?@?@????\\?&gc#{?N[Q??+?+???!@H  B`C1-2 DE@` `  P8! and elsewhere.L cer$@ @ ??t@?????PE???!@H @ B`C` DE@RX O@ @ L cer$XXD,l?@?x@????????!@H XB`DE@RXPVOXXB`C deqe D 貈 X|.  ne DE@f>>XGdeqe"* Z>!88ӢHX24`^{ 3 over 2 `} ``times`` 8`^{-` 2 over 3 `} `` [G ne)HB` C `onta 2 D E @RXG onta)B`DE@RXB`C00 D#E@<$RXB`DE@|*RXB`DE@0RXB`DE@6RXB`DE@<=RXB`DE@|CRXB`DE@IRXB`DE@RXcB`DE@@RXcB` C  osg DE@ RXcG osg@ji/Ţ*V%ILcer$cer$d ??B*B*?q4YL????!AW% H  B ` (C@key point D*E@RXO ddB`DE@RXcB`DE@RXcB`DE@@RXcB`DE@%RXcB`DE@+RXcB`DE@2RXcB`DE@@8RXcB`DE@>RXcB`DE@DRXcB`DE@KRXcB`DE@@QRXcB`DE@WRXcB`DE@]RXcB`DE@dRXcB`DE@@jRXcB`DE@pRXcB`DE@vRXcB`DE@}RXcB`DE@@RXcB`DE@RXcB`DE@RXcB` C@@ǀtǔ.... D+ +E@RXcB`DE@@RXcB`DE@RXcB`DE@RXcB`DE@RXcB`DE@@RXcB`DE@RXcB`DE@RXcB`DE@RXcB`DE@@RXcB`DE@RXcB`DE@RXcB`DE@RXcB`DE@@RXcB`DE@RXcB`DE@RXcB`DE@RXcB`DE@@RXcB` C@@ǀtǔ.... D+ +E@RXcB`DE@ RXcB`DE@RXcB`DE@@RXcB`DE@RXcB`DE@%RXcB`DE@,RXcB`DE@@2RXcB`DE@8RXcB`DE@>RXcB`DE@ERXcB`DE@RXB` C  osg DE@@ XG osgCjZ NHL` noc$noc$Z Z - ??????cer$cer$cer$NL cer$6$ ?l@?????~?0ii???Kr7KsHOL cer$/OO n??x@????0}4?o????!@uHH  B`C`8 DE@RXOOOOOL cer$ Y "  ?B@?0z@????B;6?h~????!@wHH " B`C1 - 1 DE@OY Y B`( CL 0  3@ D P tt?  ne lbt D&ERXM"M"(XG ne)HB` C `onta c$ D E @RXG onta)G lbt"*4r $HMH :$:$B`D $E @x #H ( (B` C 0 0 0 D E @RX'H T%T%B`D $E @x<$H :$:$B`C *Ʌ* D $E @x #H T%T%B`D $E @x<$H 3M 3B`D E @RXH mM mB` !C 11. deqe D E @eeXlG deqe#* e%88HXroot 4 of 16 =-2XEquation Version 60B` "C 41. deqe D E @eeXlG deqe#* e&88HXroot 3 of -27 = -3XEquation Version 60B`#C 71. deqe X @ deqe t. D E @z RXlG deqe#* (88HX9VEquation Version 60G deqe#* '88HX+-3VEquation Version 60B`$C 91. deqe X @ deqe t. D E @ccXlG deqe#* c)88HX@root4XEquation Version 60G deqe#* c*88HX@root2XEquation Version 60H 3M 3B`D E @RXH 4r4rB`D $E @xqB`CC``$ 11, 41 h   a$ 11, 71 h   b$ 11, 91 DE@}@RXB`%Cc$ 41, 71 h   d$ 41, 91 DE@FRXB`DE@LRXB`DE@=SRXB`DE@}YRXB`DE@_RXB`DE@eRXB`DE@=lRXB` C  osg DE@}r XG osgCjZ NHL` noc$noc$Z Z - ??????cer$cer$cer$ NL cer$6$ ?l@?????~?0ii???Kr7KsHOL cer$/OO n??x@????0}4?o????!@uHH  B`C`8 DE@RXOOOOOL cer$ Y "  ?B@?0z@????B;6?h~????!@wHH " B`C1 - 2 DE@OY Y B`C deqe D 貈 X|. ne DE@ՁFF XGdeqe"* F"88ԢHX M{left{ {left(9 over 16 `right)`}^{-` 3 over 4 `} `right}`}^{ 2 over 3 `} `` QG ne)HB`C @onta deqe D E @::XG onta)G deqe"* :#88բHX 4over3 `` CB`DE@sRXB`DE@RXB`DE@RXB`DE@3RXB`DE@sRXB`DE@RXB` %C  osg DE@ G osgCjZ NHL` noc$noc$Z Z - ??????cer$cer$cer$ NL cer$6$ ?l@?????~?0ii???Kr7KsHOL cer$/OO n??x@????0}4?o????!@uHH  B`C`8 DE@RXOOOOOL cer$ Y "  ?B@?0z@????B;6?h~????!@wHH " B`C1 - 3 DE@OY Y B`[%C` deqe @ deqe ǬX t, deqe X |X ǬX +ǔ deqe t. deqe X D lX$.  ne DE ?y88Gdeqe"* +880HX2`^17 ``times`` 5`^20 `` VGdeqe"* 8,881HXn`` TGdeqe"* -882HX`7` ^{1998} ``XEquation Version 60Gdeqe"* 8.883HXa`` TGdeqe"*  8/884HX n`` + ``a`` TG ne)HB` C onta 29 D E @RXG onta)B`DE@RXB`DE@RXB`DE@RXB`DE@YRXB`DE@RXB`DE@RXB` C  osg DE@ XG osgCjZ NHL` noc$noc$Z Z - ??????cer$cer$cer$ NL cer$6$ ?l@?????~?0ii???Kr7KsHOL cer$/OO n??x@????0}4?o????!@uHH  B`C`8 DE@RXOOOOOL cer$ Y "  ?B@?0z@????B;6?h~????!@wHH " B`C1 - 4 DE@OY Y B`C C`8  deqe X deqe | L, deqe X Ĭ\ 3@ @?  ne DEqvX\XGdeqe#* s88HXA,~B,~CVEquation Version 60Gdeqe#* 0t88HX 3A= 2^{1 over 2} ,~~B= 3^{1 over 3} ,~~C= 4^{1over4}]Equation Version 60Gdeqe#* u88HXA,~B,~CVEquation Version 60G ne)HB` C `onta b$ D E @RXG onta)B`DE@RXB`7 $C`$ deqe  a$ deqe  b$ deqe z  DE@O%RXGdeqe#* ev88HX@B>A>CVEquation Version 60Gdeqe#* ew88HX@C>A>BVEquation Version 60Gdeqe#* x88HX@B>A=CVEquation Version 60B` $Cc$ deqe  d$ deqe DE@+RXGdeqe#* y88¨HX@B=A>CVEquation Version 60Gdeqe#* ez88èHX@B>C>AVEquation Version 60B`$DE@1RXB`$DE@8RXB`$DE@O>RXB`$DE@DRXB`$DE@JRXB`$DE@QRXB`$DE@OWRXB`$DE@RXcB` C@@ǀtǔ.... D+ +E@@RXcB`D+E@ RXcB`D+E@RXcB`D+E@RXcB`D+E@@RXcB`D+E@%RXcB`$DE@+RXcB`$DE@2RXcB`$DE@@8RXcB`$DE@>RXcB`$DE@DRXcB`$DE@KRXcB`$DE@@QRXcB`$DE@WRXcB`$DE@]RXcB`$DE@dRXcB`$DE@@jRXcB` C@@ǀtǔ.... D+ +E@pRXcB`$DE@vRXcB`$DE@}RXcB`$DE@@RXcB`$DE@RXcB`$DE@RXcB`$DE@RXcB`$DE@@RXcB`$DE@RXcB`$DE@RXcB`$DE@RXcB`$DE@@RXcB` C@@ǀtǔ.... D+ +E@RXcB`$DE@RXcB`$DE@RXcB`$DE@@RXcB`$DE@RXcB`$DE@RXcB`$DE@RXcB`$DE@@RXcB`$DE@RXcB`$DE@RXcB`$DE@RXcB` C@@ǀtǔ.... D+ +E@@RXcB`D+E@RXcB`$DE@ RXcB`$DE@RXcB`$DE@@RXcB`$DE@RXcB`$DE@%RXcB`$DE@,RXcB`$DE@@2RXcB`$DE@8RXcB`$DE@>RXcB`$DE@ERXcB`$DE@@KRXcB`$DE@QRXcB`DE@WRXcB`C  osg osg DE XRXcG osgCj JGLcer$cer$Xaq w?"?V@ @ 0~1!@ CE ?u <V??7AJH  B `C u  - 2. X Ĭ DE@@@PO aqaqG osgCj JGL` noc$noc$ aqB ??@ @&r?(F%0.}??noc$cer$cer$JL noc$ @&r???\\?c#{???cip$lle$Lcip$ @&r???\\?c#{???V_? ` `???Uȯȯ  ȯ Llle$ @&r???\\?&gc#{?N[Q??+?+???!@H B`C1-1 DE` ` ` ` ` P8! and elsewhere.L cer$ @&r????PE???!@H B`C` DERX@RXO@ @ L cer$ @&r???????!@H B`DE@RXOXXB`C dloc osg DE@RXÃG`dloc0F5G osg@j],,,L!HLcer$cer$g&,L&??7C!?F]tE???JJH  ,LB `C X Ĭ DE@xxHIO gg&&B`DE@@RXÃB`DE@ RXÃB` C  osg DE@RXÃG osg@jg=rQ#HLcer$cer$ZP,rQ9@(??Tod?J???AJH  nrB `61C 1. deqe | L, XX  deqe X  deqe X D./E@88,<~Gdeqe"* 8 88HX a``>``0`` TGdeqe"* 888HXm`` TGdeqe"* 888HXn`` TB ` 0C t$ deqe  u$ deqe D/E@d>>,<~Gdeqe"* >88¢HX*a`^{m over n `} ``=``^n root{a`^m ``} `` [Gdeqe"* >88âHX&a`^{1 over n `} ``=``^n root{a``} `` [B `0D/E@ R,<~B `$0C 2. deqe t, deqe t X | L, D/E@88,<~Gdeqe"* 888ĢHX a``>``0``,``b``>``0`` TGdeqe"*  888ŢHX m``,``n`` TB ` 0C t$ deqe D/E@F ,<~Gdeqe"* A88ƢHX(`^n`sqrt{a`} ``)`^n`=`a` VB ` 0C u$ deqe D/E@(>>,<~Gdeqe"* 6>88ǢHX N`^n root{a``} ``^n root{b``} ``=``^n root{ab``} ``=``(a``b`)`^{1over n `} `` [B ` 0C v$ deqe D/E@&6 6 ,<~Gdeqe"* 6 88ȢHX<{`^n`sqrt{a``}}over{`^n`sqrt{b``}}`=``^n`sqrt{a over b``}` AB ` 0C w$ deqe D/E@0>>,<~Gdeqe"* />88ɢHX D(`^n root{a``} `)`^m ``=``^n root{a`^m ``} ``=``a`^{m over n `} `` [B ` 0C x$ deqe (, deqe ) D/E@^9h,<~Gdeqe"* P"88ʢHX@0`^n`sqrt{a`^m`}`=``^{n``p`}sqrt{a`^{m``p`} `}` WGdeqe"* j888ˢHXp`` TB ` 0C y$ deqe D/E@?>>,<~Gdeqe"* 3>88̢HX N`^m sqrt{`^n`sqrt{a``}``}``=``^{m``n`}sqrt{a``}``=``a`^{1 over {m``n`} `} `` [B `1D.E@HR,<~O ZZP,P,B`DE@RXÃB`DE@@RXÃB`DE@%RXÃB`DE@+RXÃB`DE@2RXÃB`DE@@8RXÃB`DE@>RXÃB`DE@DRXÃB`DE@KRXÃB`DE@@QRXÃB`DE@WRXÃB`DE@]RXÃB`DE@dRXÃB`DE@@jRXÃB`2C@ osg deqe D 貈 X|. (, deqe )  ne D4EpgXà PXÃG osgCjq JGL` noc$noc$qq 8 ??????noc$cer$cer$JL noc$7zq 8 ?K@?????\\?c#{???cip$lle$Lcip$7ȯ z$t= ?K@?????\\?c#{???V_? ` `???Uȯȯ  ȯ Llle$I 8!8! \\?@?@????\\?&gc#{?N[Q??+?+???!@H  B`C2-1 DE@` `  P8! and elsewhere.L cer$@ @ ??t@?????PE???!@H @ B`C` DE@RX O@ @ L cer$XXD,l?@?x@????????!@H XB`DE@RXPVOXXGdeqe"* : L88jHX`root { { `^3 root{a``} `} over { `^4 root{a``} `} ``} ``times``^3 root { { `^4 root{a``} `} over { root{a``} `} ``} ``times` `^4 root { {root{a``} `} over { `^3 root{a``} `} ``} `` BGdeqe"* 8M88kHX a``>``0`` TG ne)HB` C `onta 1 D E @RXG onta)B`DE@ߐRXÃB`C00 D#E@RXÃB`DE@_RXÃB`DE@RXÃB`DE@ߩRXÃB`DE@RXÃB`DE@_RXÃB`DE@RXÃB`DE@RXÃB`DE@RXÃB`DE@_RXÃB`0C osg deqe | L, deqe X D lX|. ne D5EgXÃ6 6 XÃG osgCjq JGL` noc$noc$qq 8 ??????noc$cer$cer$JL noc$7zq 8 ?K@?????\\?c#{???cip$lle$Lcip$7ȯ z$t= ?K@?????\\?c#{???V_? ` `???Uȯȯ  ȯ Llle$I 8!8! \\?@?@????\\?&gc#{?N[Q??+?+???!@H  B`C2-2 DE@` `  P8! and elsewhere.L cer$@ @ ??t@?????PE???!@H @ B`C` DE@RX O@ @ L cer$XXD,l?@?x@????????!@H XB`DE@RXPVOXXGdeqe"* N88lHXa`^{2`x`} ``=``3`` VGdeqe"* 6 O88mHX;{a`^x ``-``a`^-x `} over {a`^{3`x`} ``+``a^{-`3`x`} `} `` AG ne)HB`C @onta deqe D E @::XG onta)G deqe"* :P88nHX3 over 14 `` CB`DE@RXÃB`C00 D#E@RXÃB`DE@$RXÃB`DE@dRXÃB`DE@RXÃB`DE@RXÃB`DE@$RXÃB`DE@d!RXÃB`DE@'RXÃB`DE@-RXÃB`DE@$4RXÃB`DE@d:RXÃB`DE@@RXÃB`DE@FRXÃB`DE@RXcB` C  osg DE@@--'XcG osgCjq-Z%ILcer$cer$oqq-8??D?_T> @D@??!A[% H  qB `C ز TIP D**E@RXoB `"&C@1. deqe 1Xt m deqe \  D/E@@>>oGdeqe"* 88\%IX a`^x ``=``b`` VGdeqe"* f>88]%IXa``=``b`^{1 over x `} `` [B `&D/E@FRoB `-&C2. deqe | ` L, deqe D tǩX <\ XX D/E o$tRoGdeqe"* m88^%IXa`^x ``+``a`^{-`x`} ``=``k`` VGdeqe"* .88_%IX"a`^x ``times``a`^{-`x`} ``=``1`` VB `&D/E@$RoB `#&C`3. 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C c$ deqe    d$ deqe DE@.<<XÃGdeqe"* Z<q88HX2^20 VGdeqe"* Z<r88HX2^24 VB`DE@6RXÃB`DE@<RXÃB`DE@ CRXÃB`DE@LIRXÃB`DE@ORXÃB`DE@URXÃB`DE@RXcB`DE@@RXcB` C@@ǀtǔ.... D+ +E@ RXcB`D+E@RXcB`D+E@RXcB`D+E@@RXcB`D+E@%RXcB`D+E@+RXcB`D+E@2RXcB`D+E@@8RXcB`D+E@>RXcB`D+E@DRXcB`D+E@KRXcB`D+E@@QRXcB` C@@ǀtǔ.... D+ +E@WRXcB`D+E@]RXcB`D+E@dRXcB`D+E@@jRXcB`D+E@pRXcB`D+E@vRXcB`D+E@}RXcB`D+E@@RXcB`D+E@RXcB`D+E@RXcB`D+E@RXcB`D+E@@RXcB` C@@ǀtǔ.... D+ +E@RXcB`D+E@RXcB`D+E@RXcB`D+E@@RXcB`D+E@RXcB`D+E@RXcB`D+E@RXcB`D+E@@RXcB`D+E@RXcB`D+E@RXcB`D+E@RXcB`D+E@@RXcB`D+E@RXcB` C@@ǀtǔ.... 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DE@djj,<~Gdeqe"* $(j}883HX /a^x`=`b``````` LRARROW ``````````x`=`log_a`b` KB `0DE@ R,<~B `0C2. deqe X$t, DE@,<~Gdeqe"* q ~884HXlog`_a ``b`` HB `0C` `$  pt6" deqe DE@&88,<~Gdeqe"* 8885HX a``>``0``,~a``ne``1`` TB `0C a$  pt6" deqe DE@88,<~Gdeqe"* 8886HX b``>``0`` TB `0DE@!R,<~B ` 0C3. deqe t deqe t | L, D/E@'88,<~Gdeqe"* *8887HX@(a`>`0`,```a`neq`1`,```x`>`0`,```y`>`0` TGdeqe"* q8888HXm` TB `0C@ `$ deqe , deqe D/E@f,,<~Gdeqe"* 889HXlog_a`1`=`0` HGdeqe"* Z88:HX log_a`a`=`1 HB `0C a$ deqe D/E@~2,<~Gdeqe"* m/88;HX`)log_a `x``y``=``log_a``x``+``log_a``y`` HB `0C b$ deqe D/E@8::,<~Gdeqe"* .:88HX.log_{a`^m `} `b`^n ``=`` n over m `log_a``b` CGdeqe"*  888?HX m``,``n`` TB `0D/E@QR,<~B `0C`4. deqe P 1t D̲ | L, D/E@ W88,<~Gdeqe"* 888@HX`a``,``b``,``c`` TB `0C `$ deqe , deqe D/E@p\ *,<~Gdeqe"*  88AHX$log_a`b`=`{log_c`b}over{log_c`a`}` @Gdeqe"* 88BHXlog_a`b`=`1over{log_b`a`}` >B `0C  a$ deqe , deqe D/E@>g""w,<~Gdeqe"* W"88CHX@a`^{log_a`b`} ``=``b`` WGdeqe"* "88DHX`)a`^{log_c ``b`} ``=``b`^{log_c ``a`} `` WO ZZP,P,B`DE@d$RXÃB`DE@*RXÃB`DE@0RXÃB`DE@$7RXÃB`DE@d=RXÃB`DE@CRXÃB`DE@IRXÃB`DE@$PRXÃB`DE@dVRXÃB`DE@\RXÃB`DE@bRXÃB`DE@$iRXÃB`DE@doRXÃB`DE@uRXÃB`DE@{RXÃB`DE@$RXÃB`DE@dRXÃB`DE@RXÃB`DE@RXÃB`DE@$RXÃB`6C osg deqe Xij] XՔ deqe X X ǔ?  ne D3,EdgXÃ۵UU XÃG osgCjqNHL` noc$noc$qq 8 ??????noc$cer$cer$NL noc$7zq 8 ?K@?????\\?c#{???cip$lle$Lcip$7ȯ z$t= ?K@?????\\?c#{???V_? ` `???Uȯȯ  ȯ Llle$I 8!8! \\?@?@????\\?&gc#{?N[Q??+?+???!@H  B`C3-1 DE@` `  P8! and elsewhere.L cer$@ @ ??t@?????PE???!@H @ B`C` DE@RX O@ @ L cer$XXD,l?@?x@????????!@H XB`DE@RXPVOXXGdeqe#* )U881NHXlog _{(x-3)} (-x ^{2} +5x-4)`LEquation Version 60Gdeqe#* 882NHXx`VEquation Version 60G ne )HB` C `onta d$ D E @RXG onta )B`7 $C`$ deqe  a$ deqe  b$ deqe : DE@RXÃGdeqe#* 883NHX@1<x<2VEquation Version 60Gdeqe#* 884NHX@1<x<3VEquation Version 60Gdeqe#* 885NHX@2<x<3VEquation Version 60B` $Cc$ deqe  d$ deqe DE@RXÃGdeqe#* 886NHX@2<x<4VEquation Version 60Gdeqe#* 887NHX@3<x<4VEquation Version 60B`DE@RXÃB`C00 D#E@HRXÃB`DE@RXÃB`DE@RXÃB`DE@RXÃB`DE@HRXÃB`DE@RXÃB`DE@RXÃB` C osg deqe X @?  ne DEgXÃXÃG osgCjqNHL` noc$noc$qq 8 ??????noc$cer$cer$NL noc$7zq 8 ?K@?????\\?c#{???cip$lle$Lcip$7ȯ z$t= ?K@?????\\?c#{???V_? ` `???Uȯȯ  ȯ Llle$I 8!8! \\?@?@????\\?&gc#{?N[Q??+?+???!@H  B`C3-2 DE@` `  P8! and elsewhere.L cer$@ @ ??t@?????PE???!@H @ B`C` DE@RX O@ @ L cer$XXD,l?@?x@????????!@H XB`DE@RXPVOXXGdeqe"* r!88BNHX 73 ^{2log _{3} ``4+log _{3} ``5-3`log _{3} ``2}``` YEquation Version 60G ne)HB` C `onta c$ D E @RXG onta)B`7 $C`$ deqe S a$ deqe S b$ deqe  DE@RXÃGdeqe"* 88CNHX3 VEquation Version 60Gdeqe"* 88DNHX4 VEquation Version 60Gdeqe"* 88ENHX5 VEquation Version 60B` $Cc$ deqe F d$ deqe DE@6RXÃGdeqe"* 88FNHX 10 VEquation Version 60Gdeqe"* 88GNHX 11 VEquation Version 60B`DE@v$RXÃB`C00 D#E@*RXÃB`DE@0RXÃB`DE@67RXÃB`DE@v=RXÃB`DE@CRXÃB`DE@IRXÃB`DE@6PRXÃB`DE@vVRXÃB`DE@RXcB`DE@@RXcB`DE@ RXcB` C  osg DE@RXcG osg@ji/Ţ-c%ILcer$cer$d ??B*B*?q4YL????!Ad% H  B ` (C@key point D*E@RXO ddB`DE@RXcB`DE@@RXcB`DE@%RXcB`DE@+RXcB`DE@2RXcB`DE@@8RXcB`DE@>RXcB`DE@DRXcB`DE@KRXcB`DE@@QRXcB`DE@WRXcB`DE@]RXcB`DE@dRXcB`DE@@jRXcB`DE@pRXcB`DE@vRXcB`DE@}RXcB`DE@@RXcB`DE@RXcB`DE@RXcB`DE@RXcB`DE@@RXcB` C@@ǀtǔ.... 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D+ +E@@RXcB`DE@RXcB`DE@ RXcB`DE@RXcB`DE@@RXcB`DE@RXcB`DE@%RXcB`DE@,RXcB`DE@@2RXcB`DE@8RXcB`DE@>RXcB`DE@ERXcB`DE@@KRXcB`DE@QRXcB`DE@WRXcB`DE@RXÃB` C  osg DE@@ XÃG osgCjZ NNHL` noc$noc$Z Z - ??????cer$cer$cer$ONL cer$6$ ?l@?????~?0ii???Kr7KsHOL cer$/OO n??x@????0}4?o????!@uHH  B`C`8 DE@RXOOOOOL cer$ Y "  ?B@?0z@????B;6?h~????!@wHH " B`C3 - 1 DE@OY Y B`C deqe X @?  ne DE@t t nXÃGdeqe#* D%t 88XNHX ;LEFT ( log_4`12 -log_4 ` { 3} over {4 } RIGHT )^{log_2`5}``DEquation Version 60G ne)HB` C `onta a$ D E @RXG onta)B`8 $C`$ deqe  a$ deqe  b$ deqe  DE@d!RXÃGdeqe#* 88YNHX4VEquation Version 60Gdeqe#* 88ZNHX5VEquation Version 60Gdeqe#* 88[NHX8VEquation Version 60B` $Cc$ deqe  d$ deqe DE@'RXÃGdeqe#* 88\NHX16VEquation Version 60Gdeqe#* 88]NHX25VEquation Version 60B`DE@-RXÃB`DE@$4RXÃB`DE@d:RXÃB`DE@@RXÃB`D'E@FRXÃB`D'E@$MRXÃB` C  osg D'E@dS XÃG osgCjZ PNHL` noc$noc$Z Z - ??????cer$cer$cer$QNL cer$6$ ?l@?????~?0ii???Kr7KsHOL cer$/OO n??x@????0}4?o????!@uHH  B`C`8 DE@RXOOOOOL cer$ Y "  ?B@?0z@????B;6?h~????!@wHH " B`C3 - 2 DE@OY Y B`C deqe , deqe | L, D'E@bxxXÃGdeqe"* Qx88ZHX log_3 ``a=x HGdeqe"* Qx88ZHX log_3 ``b=y HB`:C@ deqe | ̹qXՔ  deqe X i deqe X D lX|.  ne D'Ej XÃvXÃGdeqe"* P 88ZHX Blog_3 ``a b^2 +log_root3 ``a^2 over b - 2log_3 ``a^2 b^2 =px+qy` EGdeqe"* 88ZHX p,``q TGdeqe"* @ 88ZHXp+q` TG ne)HB`  C onta -3 D E @G onta)B`D'E@}RXÃB`D'E@܃RXÃB`D'E@RXÃB`D'E@\RXÃB`D'E@RXÃB`D'E@ܜRXÃB` C  osg D'E@ XÃG osgCjZ RNHL` noc$noc$Z Z - ??????cer$cer$cer$SNL cer$6$ ?l@?????~?0ii???Kr7KsHOL cer$/OO n??x@????0}4?o????!@uHH  B`C`8 DE@RXOOOOOL cer$ Y "  ?B@?0z@????B;6?h~????!@wHH " B`C3 - 3 DE@OY Y B`CC`t() deqe X P t deqe , deqe | L, deqe X @?  ne D':-EtxxXÃ,DxxÃGdeqe"* <88ZHX x^2 -5x+1=0 VGdeqe"* x88ZHX {log_2}a HGdeqe"* x88ZHX {log_2}b HGdeqe"* [x88ZHX{log_a}b {log_b}a HG ne)HB`  C `onta a$ D E @G onta)B`0C`$ 21 \  a$ 23   b$ 25 D'E@RXÃB`C c$ 27   d$ 29 D'E@RXÃB`D'E@ RXÃB`D'E@LRXÃB`D'E@RXÃB`D'E@RXÃB`D'E@ RXÃB`D'E@LRXÃB` C  osg D'E@ XÃG osgCjZ TNHL` noc$noc$Z Z - ??????cer$cer$cer$UNL cer$6$ ?l@?????~?0ii???Kr7KsHOL cer$/OO n??x@????0}4?o????!@uHH  B`C`8 DE@RXOOOOOL cer$ Y "  ?B@?0z@????B;6?h~????!@wHH " B`C3 - 4 DE@OY Y B`C C`8  deqe X deqe | L, deqe X Ĭ\ 3@ @?  ne D':'E\XÃ$ \XÃ7"RXÃGdeqe#* 88ZHX A,```B,```CVEquation Version 60Gdeqe#* M:88ZHX 4A=log _{2} 3,``````B=log _{3} 2,`````C=log _{4} 8HEquation Version 60Gdeqe#* 88ZHX A,```B,```CVEquation Version 60G ne)HB` C `onta a$ D E @RXG onta)B`% C`$ deqe  a$ deqe q  D'E@bRXÃGdeqe#* e88ZHX@A>B>CVEquation Version 60Gdeqe#* e88ZHX@A>C>BVEquation Version 60B` Cb$ deqe  c$ deqe D'E@RXÃGdeqe#* e88ZHX@B>A>CVEquation Version 60Gdeqe#* e88ZHX@B>C>AVEquation Version 60B` C`d$ deqe D'E@#RXÃGdeqe#* e88 ZHX@C>A>BVEquation Version 60B`D'E@"*RXÃB`D'E@b0RXÃB`DE@6RXÃB`DE@<RXÃB`DE@"CRXÃB`DE@bIRXÃB`DE@ORXÃB`DE@URXÃB`DE@RXcB` C@@ǀtǔ.... D+ +E@@RXcB`D+E@ RXcB`D+E@RXcB`D+E@RXcB`D+E@@RXcB`D+E@%RXcB`D+E@+RXcB`D+E@2RXcB`D+E@@8RXcB`D+E@>RXcB`D+E@DRXcB`D+E@KRXcB`D+E@@QRXcB` C@@ǀtǔ.... D+ +E@WRXcB`D+E@]RXcB`D+E@dRXcB`D+E@@jRXcB`D+E@pRXcB`D+E@vRXcB`D+E@}RXcB`D+E@@RXcB`D+E@RXcB`D+E@RXcB`D+E@RXcB`D+E@@RXcB`D+E@RXcB` C@@ǀtǔ.... D+ +E@RXcB`D+E@RXcB`D+E@@RXcB`D+E@RXcB`D+E@RXcB`D+E@RXcB`D+E@@RXcB`D+E@RXcB`D+E@RXcB`D+E@RXcB`D+E@@RXcB`D+E@RXcB`D+E@RXcB` C@@ǀtǔ.... D+ +E@RXcB`D+E@@RXcB`D+E@RXcB`D+E@ RXcB`D+E@RXcB`D+E@@RXcB`D+E@RXcB`D+E@%RXcB`D+E@,RXcB`D+E@@2RXcB`D+E@8RXcB`D+E@>RXcB`D+E@ERXcB`D+E@@KRXcB`D+E@QRXcB`DE@WRXcB` C  osg DE@ XÃG osgCj ZHLcer$cer$Xaq w?"?V@ @ 0~1!@ CE ?u <V??7AZH  B `Cu  - 4. \X \@  DE@@@PO aqaqB` C  osg DE@d RXÃG osg@z],,,L'-ZHLcer$cer$g&,L&??7C!?F]tE???J.ZH  ,LB `C \X \@  DE@xxHIO gg&&B`DE@RXÃB`DE@RXÃB` C  osg DE@$RXÃG osg@js=ri^(/ZHLcer$cer$ZP,ri^9@4/??Tod?_ @??A0ZH  nrB ` 2C`1. \@  D&E@R<~B `)2C  deqe (, deqe @ , deqe ) D&E@R<~Gdeqe"* 88ZHX!log`` rm N``= it ``n``+`` alpha `VEquation Version 60Gdeqe#* 88ZHXnVEquation Version 60Gdeqe#* 88ZHX 0 LEQ alpha PREC 1VEquation Version 60B `2C deqe , deqe D&E@ R<~Gdeqe#* 88ZHX n=[log`N]VEquation Version 60Gdeqe#* 88ZHX@alpha =log``N-[log`N]VEquation Version 60B `2D&E@R<~B `2C2. \X 1 (Ǭ 8) D&E@pR<~B `2C` t$ deqe ! deqe Ǭ D&E@LR<~Gdeqe"* B88ZHX log`` rm N``=``n`.` CDOTS ``VEquation Version 60Gdeqe"* 88ZHX  (n``+``1`)``VEquation Version 60B `42C u$ deqe ! deqe Ǭ L<\ 0 t D̲ + (. D&E(#R<~/)R<~Gdeqe"* 88ZHX`log`` rm N``=``-`n`.` CDOTS ``VEquation Version 60Gdeqe"* 88ZHX  (n``+``1`)``VEquation Version 60B `2D&E@.R<~B ` 2C`3. X 1 D&E@4R<~B `2C t$   @@??!A[H  cB `C ز TIP D*E@RXPaB `D*E@@RXPaB ` &C1. A| deqe ɕD дD LX Ǭ D/E@ RPaGdeqe#*  88[HXpVEquation Version 60B `&C` : ( deqe X \) deqe D/E@066RPaGdeqe#* 688[HX log _{p} AGEquation Version 60Gdeqe#* 88[HX+1VEquation Version 60B `&D/E@.RPaB `&C@2. deqe X \ deqe D/E@RPaGdeqe#* 88[HX@log`AVEquation Version 60Gdeqe#* 88[HXnVEquation Version 60B ` &C ! deqe D/E@PaGdeqe#* @88[HX In LEQ logA PREC (a+1)`````` LRARROW ``````10 ^{n} LEQ A PREC 10 ^{n+1}XEquation Version 60O qqB`DE@Y@RXcB`DE@FRXcB`DE@LRXcB`DE@SRXcB`DE@YYRXcB`DE@_RXcB`DE@eRXcB`DE@lRXcB`DE@YrRXcB`DE@xRXcB`DE@~RXcB`DE@RXcB` C@@ǀtǔ.... D+ +E@YRXcB`DE@RXcB`DE@ٗRXcB`DE@RXcB`DE@YRXcB`DE@RXcB`DE@ٰRXcB`DE@RXcB`DE@YRXcB`DE@RXcB`DE@RXcB`DE@RXcB` C@@ǀtǔ.... D+ +E@YRXcB`DE@RXcB`DE@RXcB`DE@RXcB`DE@YRXcB`DE@RXcB`DE@RXcB`DE@RXcB`DE@YRXcB`DE@RXcB` C@@ǀtǔ.... D+ +E@RXcB`DE@RXcB`DE@Y!RXcB`DE@'RXcB`DE@-RXcB`DE@4RXcB`DE@Y:RXcB`DE@@RXcB`DE@FRXcB`DE@MRXcB`DE@YSRXcB`DE@RXÃB` C  osg DE@@ XÃG osgCjZ UZHL` noc$noc$Z Z - ??????cer$cer$cer$VZL cer$6$ ?l@?????~?0ii???Kr7KsHOL cer$/OO n??x@????0}4?o????!@uHH  B`C`8 DE@RXOOOOOL cer$ Y "  ?B@?0z@????B;6?h~????!@wHH " B`C4 - 1 DE@OY Y B`g C P  deqe X deqe @ deqe ǬX , deqe @ deqe ǬX | L, deqe @ ǬX x?  ne DE XÃS{RXÃGdeqe#* A88ZHXa,bVEquation Version 60Gdeqe#* 88ZHX a^10YEquation Version 60Gdeqe#* 88ZHX16VEquation Version 60Gdeqe#* 88ZHX b^10YEquation Version 60Gdeqe#* 88ZHX22VEquation Version 60Gdeqe#* 88ZHXa^3 b^5YEquation Version 60G ne)HB` C `onta b$ D E @RXG onta)B`<  C`$ deqe   a$ deqe   b$ deqe DE@"RXÃGdeqe#* 88ZHX14VEquation Version 60Gdeqe#* 88ZHX15VEquation Version 60Gdeqe#* 88ZHX16VEquation Version 60B`#  C`c$ deqe   d$ deqe DE@(RXÃGdeqe#* 88ZHX17VEquation Version 60Gdeqe#* 88ZHX18VEquation Version 60B`DE@;/RXÃB`DE@{5RXÃB`DE@;RXÃB`DE@ARXÃB` C  osg DE@;H XÃG osgCjZ _ZHL` noc$noc$Z Z - ??????cer$cer$cer$`ZL cer$6$ ?l@?????~?0ii???Kr7KsHOL cer$/OO n??x@????0}4?o????!@uHH  B`C`8 DE@RXOOOOOL cer$ Y "  ?B@?0z@????B;6?h~????!@wHH " B`C4 - 2 DE@OY Y B`LC deqe t deqe X \| deqe X | deqe | ` L, deqe X @?  ne DCEW XÃCqcRXÃGdeqe#* *88ZHXlogx= bar {1 }.3YEquation Version 60Gdeqe#*  88ZHX`logx^4YEquation Version 60Gdeqe#* | 88ZHXa,~log{1 over{x^2}=Equation Version 60Gdeqe#* 88ZHXbVEquation Version 60Gdeqe#* 88ZHXa+bVEquation Version 60G ne)HB` C onta a$ D  E @RXG onta)B`7 $C`$ deqe  a$ deqe  b$ deqe  DE@iRXÃGdeqe#* 88ZHX -2.8VEquation Version 60Gdeqe#* 88ZHX -2.6VEquation Version 60Gdeqe#* 88ZHX -2.4VEquation Version 60B` $Cc$ deqe  d$ deqe DE@oRXÃGdeqe#* 88ZHX -2.2VEquation Version 60Gdeqe#* 88ZHX -2.0VEquation Version 60B`DE@1vRXÃB`DE@q|RXÃB`DE@RXÃB`DE@RXÃB`DE@1RXÃB` C  osg DE@q XÃG osgCjZ fZHL` noc$noc$Z Z - ??????cer$cer$cer$gZL cer$6$ ?l@?????~?0ii???Kr7KsHOL cer$/OO n??x@????0}4?o????!@uHH  B`C`8 DE@RXOOOOOL cer$ Y "  ?B@?0z@????B;6?h~????!@wHH " B`C4 - 3 DE@OY Y B`GC deqe X deqe X @ deqe X  D L, deqe X @?  ne D>EɤXÃ;RXÃGdeqe#* 88ZHX  10x100`x~xVEquation Version 60Gdeqe#* 88ZHX logxVEquation Version 60Gdeqe#* 88ZHX@ log {1 over xBEquation Version 60Gdeqe#* X88ZHXxVEquation Version 60G ne)HB` C `onta d$ D E @RXG onta)B`7 $C`$ deqe  a$ deqe  b$ deqe  DE@+XÃGdeqe#* e88ZHX2 root {3} of {10}XEquation Version 60Gdeqe#* 88ZHX{2 over3}sqrt {10 }BEquation Version 60Gdeqe#* e88ZHX@ {2}sqrt {10 }XEquation Version 60B` $Cc$ deqe  d$ deqe DE@MeeXÃGdeqe#* e88ZHX 5sqrt {10 }XEquation Version 60Gdeqe#*  e88ZHX  10sqrt {10 }XEquation Version 60B`DE@ RXÃB`DE@JRXÃB`DE@RXÃB`DE@RXÃB`DE@ RXÃB` C  osg DE@J XÃG osgCjZ lZHL` noc$noc$Z Z - ??????cer$cer$cer$mZL cer$6$ ?l@?????~?0ii???Kr7KsHOL cer$/OO n??x@????0}4?o????!@uHH  B`C`8 DE@RXOOOOOL cer$ Y "  ?B@?0z@????B;6?h~????!@wHH " B`C4 - 4 DE@OY Y B`_C  deqe X deqe X \@ |  deqe , deqe | ` L, <0> 3@ D P x @?  ne DEXÃARXÃGdeqe"* 888[HXa` ` TGdeqe"*  88[HX` loga`` TGdeqe"* 88[HX f`(a`)` TGdeqe"* x 88[HX` g`(a`)` TG ne)HB` C `onta d$ D E @RXG onta)B` +C  lbt DE@"""lWG lbt"*w?8T[HMH 22B`)D ,E @ddU42H  B`,C < 0> D E @RH 22B`)D ,E @ddU42H 22B`)D ,E @ddU42H 22B`)D ,E @ddU42H wwB` - C `11.  deqe D E @ wG deqe"* 888[HXf`(2006 )=3` TB` - C `41.  deqe D E @ wG deqe"* . 88[HXg`(2 )+ g`(6) = g`(12)+1 TB`" - C @71.  deqe tt deqe t. 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