0)1) !#$%&'%()#*)+#,-'&./() +%((#0)1)...

8
Properties of Logarithms !"#$%& x "#$ %#&'($ )* +&,$ *)-./0&1 x 2)(&-34#., )* 5-)6/%4,7 8/)43$'4,7 &'6 5)9$-,1 x :5503%&43)', )* 0)(&-34#.,1 !()"*)+& ,-. /*"#)*0$)& x "#$ %(,-1) "2 3,&) 2"*+45, 3, ;< ;< 0)( 0' 0)( 1 0)( 0' ! " " " ! ! x 0)( = > 0)( 0)( ! ! ! #$ # $ x 0)( 0)( 0)( ! ! ! # # $ $ § · ¨ ¸ © ¹ x 0)( 0)( % ! ! # % # !() 645) "2 78 "#$ 6)/+03'( 43.$ *)- & 6$5),34 $&-'3'( & ? 3'4$-$,4 %).5)/'6$6 !%%#!''( 3, @A & B$&-,1 Note: IB DP SL Mathematics Students are not required to know Rule 72. ;<,+#5) => ?,5%45,0$-1 @"1,*$0(+& x ;< F ;< 0)( AG ;1HI@IF 0)( AG A1HA 0)( F <1J<A<J | | x F 0' AG H1A;KKK 0)( AG A1HA 0' F ;1HKJAI | | !"#$%& #()# #(& * )+,-&., )/.&&0 "#$%&'( )* +,%&',-.,/0 120$3,45%6 1+& "2 #(& 3",# $34".#)+# 4."4&.#$&, "2 5"/).$#(3, $, #()# #(& 5"/).$#(3, "2 4."67%#, ).& #(& ,73, "2 #(& 5"/).$#(3,0 8$3$5).59: #(& 5"/).$#(3, "2 ;7"#$&+#, ).& #(& 6$22&.&+%&, "2 #(& 5"/).$#(3,0 x > @ 5+ * 5+ < 5+ =*>=<> 5+ ? x * 5+ 5+ * 5+ *@ *@ x * AB AB AB 5"/ CD 5"/ @ * 5"/ @ Topic 1 - Algebra (Exponents and Logarithms) 1

Upload: others

Post on 06-Aug-2020

9 views

Category:

Documents


1 download

TRANSCRIPT

Page 1: 0)1) !#$%&'%()#*)+#,-'&./() +%((#0)1) !#$%&'%(dpmathematicssl.weebly.com/uploads/1/2/...algebra_laws_of_expone… · !"#$%&'#()#'#(&'*')+,-&.,')/.&&0' '!! "#$%&'(!)*!+,%&',-.,/0!120$3,45%6!

Properties of Logarithms

!"

!"#$$%&!'(!)*%+#*,-#$!%.!"%/0*-,12$

!"#$%"&'%()#*)+#,-"'&./()+%((#0)1)

!

!"#$%&'

! "#$!%#&'($!)*!+&,$!*)-./0&1!! 2)(&-34#.,!)*!5-)6/%4,7!8/)43$'4,7!&'6!5)9$-,1!! :5503%&43)',!)*!0)(&-34#.,1!

!

!()"*)+&',-.'/*"#)*0$)&! '

! "#$!%(,-1)'"2'3,&)'2"*+45,'3,! ;<

;<

0)( 0'0)( 10)( 0'!

" ""! !

!

! 0)( = > 0)( 0)(! ! !#$ # $ !'

! 0)( 0)( 0)(! ! !# # $$

'

! 0)( 0)(%! !# % # '

'

!()'645)'"2'78'

"#$'6)/+03'(!43.$!*)-!&!6$5),34!$&-'3'(! &?!3'4$-$,4!%).5)/'6$6!!%%#!''(!3,' @A& 'B$&-,1'

'

94++,*:'

C-&5#3'(!%&0%/0&4)-,!/,/&00B!#&D$!)'0B!A!E$B,!*)-! 0)(&-34#.,7! 4#),$! 4)!+&,$!;<!&'6! 4#$!'&4/-&0! 0)(&-34#.1!")!%&0%/0&4$!0)(&-34#.,!4)!)4#$-!+&,$,7!/,$!4#$!%#&'($!)*!+&,$!*)-./0&1!!

'

;<,+#5)'=>'?,5%45,0$-1'@"1,*$0(+&'

! ;<F

;<

0)( AG ;1HI@IF0)( AG A1HA0)( F <1J<A<J

!!

! F0' AG H1A;KKK0)( AG A1HA0' F ;1HKJAI

!

!"

!"#$$%&!'(!)*%+#*,-#$!%.!"%/0*-,12$

!"#$%"&'%()#*)+#,-"'&./()+%((#0)1)

!

!"#$%&'

! "#$!%#&'($!)*!+&,$!*)-./0&1!! 2)(&-34#.,!)*!5-)6/%4,7!8/)43$'4,7!&'6!5)9$-,1!! :5503%&43)',!)*!0)(&-34#.,1!

!

!()"*)+&',-.'/*"#)*0$)&! '

! "#$!%(,-1)'"2'3,&)'2"*+45,'3,! ;<

;<

0)( 0'0)( 10)( 0'!

" ""! !

!

! 0)( = > 0)( 0)(! ! !#$ # $ !'

! 0)( 0)( 0)(! ! !# # $$

'

! 0)( 0)(%! !# % # '

'

!()'645)'"2'78'

"#$'6)/+03'(!43.$!*)-!&!6$5),34!$&-'3'(! &?!3'4$-$,4!%).5)/'6$6!!%%#!''(!3,' @A& 'B$&-,1'

'

94++,*:'

C-&5#3'(!%&0%/0&4)-,!/,/&00B!#&D$!)'0B!A!E$B,!*)-! 0)(&-34#.,7! 4#),$! 4)!+&,$!;<!&'6! 4#$!'&4/-&0! 0)(&-34#.1!")!%&0%/0&4$!0)(&-34#.,!4)!)4#$-!+&,$,7!/,$!4#$!%#&'($!)*!+&,$!*)-./0&1!!

'

;<,+#5)'=>'?,5%45,0$-1'@"1,*$0(+&'

! ;<F

;<

0)( AG ;1HI@IF0)( AG A1HA0)( F <1J<A<J

!!

! F0' AG H1A;KKK0)( AG A1HA0' F ;1HKJAI

!

!"

!"#$$%&!'(!)*%+#*,-#$!%.!"%/0*-,12$

!"#$%"&'%()#*)+#,-"'&./()+%((#0)1)

!

!"#$%&'

! "#$!%#&'($!)*!+&,$!*)-./0&1!! 2)(&-34#.,!)*!5-)6/%4,7!8/)43$'4,7!&'6!5)9$-,1!! :5503%&43)',!)*!0)(&-34#.,1!

!

!()"*)+&',-.'/*"#)*0$)&! '

! "#$!%(,-1)'"2'3,&)'2"*+45,'3,! ;<

;<

0)( 0'0)( 10)( 0'!

" ""! !

!

! 0)( = > 0)( 0)(! ! !#$ # $ !'

! 0)( 0)( 0)(! ! !# # $$

'

! 0)( 0)(%! !# % # '

'

!()'645)'"2'78'

"#$'6)/+03'(!43.$!*)-!&!6$5),34!$&-'3'(! &?!3'4$-$,4!%).5)/'6$6!!%%#!''(!3,' @A& 'B$&-,1'

'

94++,*:'

C-&5#3'(!%&0%/0&4)-,!/,/&00B!#&D$!)'0B!A!E$B,!*)-! 0)(&-34#.,7! 4#),$! 4)!+&,$!;<!&'6! 4#$!'&4/-&0! 0)(&-34#.1!")!%&0%/0&4$!0)(&-34#.,!4)!)4#$-!+&,$,7!/,$!4#$!%#&'($!)*!+&,$!*)-./0&1!!

'

;<,+#5)'=>'?,5%45,0$-1'@"1,*$0(+&'

! ;<F

;<

0)( AG ;1HI@IF0)( AG A1HA0)( F <1J<A<J

!!

! F0' AG H1A;KKK0)( AG A1HA0' F ;1HKJAI

!

Note: IB DP SL Mathematics Students are not required to know Rule 72.

!"

!"#$$%&!'(!)*%+#*,-#$!%.!"%/0*-,12$

!"#$%"&'%()#*)+#,-"'&./()+%((#0)1)

!

!"#$%&'

! "#$!%#&'($!)*!+&,$!*)-./0&1!! 2)(&-34#.,!)*!5-)6/%4,7!8/)43$'4,7!&'6!5)9$-,1!! :5503%&43)',!)*!0)(&-34#.,1!

!

!()"*)+&',-.'/*"#)*0$)&! '

! "#$!%(,-1)'"2'3,&)'2"*+45,'3,! ;<

;<

0)( 0'0)( 10)( 0'!

" ""! !

!

! 0)( = > 0)( 0)(! ! !#$ # $ !'

! 0)( 0)( 0)(! ! !# # $$

'

! 0)( 0)(%! !# % # '

'

!()'645)'"2'78'

"#$'6)/+03'(!43.$!*)-!&!6$5),34!$&-'3'(! &?!3'4$-$,4!%).5)/'6$6!!%%#!''(!3,' @A& 'B$&-,1'

'

94++,*:'

C-&5#3'(!%&0%/0&4)-,!/,/&00B!#&D$!)'0B!A!E$B,!*)-! 0)(&-34#.,7! 4#),$! 4)!+&,$!;<!&'6! 4#$!'&4/-&0! 0)(&-34#.1!")!%&0%/0&4$!0)(&-34#.,!4)!)4#$-!+&,$,7!/,$!4#$!%#&'($!)*!+&,$!*)-./0&1!!

'

;<,+#5)'=>'?,5%45,0$-1'@"1,*$0(+&'

! ;<F

;<

0)( AG ;1HI@IF0)( AG A1HA0)( F <1J<A<J

!!

! F0' AG H1A;KKK0)( AG A1HA0' F ;1HKJAI

!

!"

!"#$%&'#()#'#(&'*')+,-&.,')/.&&0' '

!!

"#$%&'(!)*!+,%&',-.,/0!120$3,45%6!

1+&' "2' #(&' 3",#' $34".#)+#' 4."4&.#$&,' "2' 5"/).$#(3,' $,' #()#' #(&' 5"/).$#(3,' "2' 4."67%#,' ).&' #(&' ,73,' "2' #(&'

5"/).$#(3,0'8$3$5).59:'#(&'5"/).$#(3,'"2';7"#$&+#,').&'#(&'6$22&.&+%&,'"2'#(&'5"/).$#(3,0'

! 5+ * 5+< 5+ =*>=<> 5+? '

!*

5+ 5+ * 5+ *@*@

'

! *

AB AB AB5"/ CD 5"/ @ *5"/ @ '

!!

"#$%&'(!7*!83,4,/0!$!932:;<4!$6!$!+;%!

E"/).$#(3,'45)9')'3)F".'."5&'$+'%)5%757,'G9'%"+H&.#$+/'4."67%#,'#"',73,0!

! < <

C C C C C C C5"/ I 5"/ I 5"/ 5"/ 5"/ I <5"/ 5"/! " ! " ! " '

!A* A

*< I< I A

5+ 5+ 5+ < I 5+ @ 5+ < I 5+ @@ @ *

!! ! ! ' '

'

'

'

"#$%&'(!=*!>5(!?;'(!2-!@)!

J(&' .75&' "2' @*' $,' )' %"+H&+$&+#' #""5' 2".' &,#$3)#$+/' #(&' 6"7G5$+/' #$3&' 2".' )+' $+H&,#3&+#' &).+$+/' $+#&.&,#'

%"34"7+6&6' )++7)5590' 8744",&' )' G)+K' 4)9,' LM' $+#&.&,#:' %"34"7+6&6' )++7)5590' N,&' #(&' .75&' "2' @*' #"'

)44."O$3)#&'#(&'6"7G5$+/'#$3&0'

+2';4,2/!

J(&'.75&'"2'@*',)9,'#()#'#(&'6"7G5$+/'#$3&'$,')44."O$3)#&59'@*

DL

'9&).,0'P2'9"7'7,&'#(&'2".375)'2".'%"34"7+6'

$+#&.&,#:'9"7'-$55',&&'#()#'#($,')44."O$3)#$"+'$,'&O%&55&+#0'

!!

! !!"

!"#$%&'#()#'#(&'*')+,-&.,')/.&&0' '

!!

"#$%&'(!)*!+,%&',-.,/0!120$3,45%6!

1+&' "2' #(&' 3",#' $34".#)+#' 4."4&.#$&,' "2' 5"/).$#(3,' $,' #()#' #(&' 5"/).$#(3,' "2' 4."67%#,' ).&' #(&' ,73,' "2' #(&'

5"/).$#(3,0'8$3$5).59:'#(&'5"/).$#(3,'"2';7"#$&+#,').&'#(&'6$22&.&+%&,'"2'#(&'5"/).$#(3,0'

! 5+ * 5+< 5+ =*>=<> 5+? '

!*

5+ 5+ * 5+ *@*@

'

! *

AB AB AB5"/ CD 5"/ @ *5"/ @ '

!!

"#$%&'(!7*!83,4,/0!$!932:;<4!$6!$!+;%!

E"/).$#(3,'45)9')'3)F".'."5&'$+'%)5%757,'G9'%"+H&.#$+/'4."67%#,'#"',73,0!

! < <

C C C C C C C5"/ I 5"/ I 5"/ 5"/ 5"/ I <5"/ 5"/! " ! " ! " '

!A* A

*< I< I A

5+ 5+ 5+ < I 5+ @ 5+ < I 5+ @@ @ *

!! ! ! ' '

'

'

'

"#$%&'(!=*!>5(!?;'(!2-!@)!

J(&' .75&' "2' @*' $,' )' %"+H&+$&+#' #""5' 2".' &,#$3)#$+/' #(&' 6"7G5$+/' #$3&' 2".' )+' $+H&,#3&+#' &).+$+/' $+#&.&,#'

%"34"7+6&6' )++7)5590' 8744",&' )' G)+K' 4)9,' LM' $+#&.&,#:' %"34"7+6&6' )++7)5590' N,&' #(&' .75&' "2' @*' #"'

)44."O$3)#&'#(&'6"7G5$+/'#$3&0'

+2';4,2/!

J(&'.75&'"2'@*',)9,'#()#'#(&'6"7G5$+/'#$3&'$,')44."O$3)#&59'@*

DL

'9&).,0'P2'9"7'7,&'#(&'2".375)'2".'%"34"7+6'

$+#&.&,#:'9"7'-$55',&&'#()#'#($,')44."O$3)#$"+'$,'&O%&55&+#0'

!!

! !

Topic 1 - Algebra (Exponents and Logarithms) 1

Page 2: 0)1) !#$%&'%()#*)+#,-'&./() +%((#0)1) !#$%&'%(dpmathematicssl.weebly.com/uploads/1/2/...algebra_laws_of_expone… · !"#$%&'#()#'#(&'*')+,-&.,')/.&&0' '!! "#$%&'(!)*!+,%&',-.,/0!120$3,45%6!

!"

!"#$%&'#()#'#(&'*')+,-&.,')/.&&0' '

!!

"#$%&'(!)*!+,%&',-.,/0!120$3,45%6!

1+&' "2' #(&' 3",#' $34".#)+#' 4."4&.#$&,' "2' 5"/).$#(3,' $,' #()#' #(&' 5"/).$#(3,' "2' 4."67%#,' ).&' #(&' ,73,' "2' #(&'

5"/).$#(3,0'8$3$5).59:'#(&'5"/).$#(3,'"2';7"#$&+#,').&'#(&'6$22&.&+%&,'"2'#(&'5"/).$#(3,0'

! 5+ * 5+< 5+ =*>=<> 5+? '

!*

5+ 5+ * 5+ *@*@

'

! *

AB AB AB5"/ CD 5"/ @ *5"/ @ '

!!

"#$%&'(!7*!83,4,/0!$!932:;<4!$6!$!+;%!

E"/).$#(3,'45)9')'3)F".'."5&'$+'%)5%757,'G9'%"+H&.#$+/'4."67%#,'#"',73,0!

! < <

C C C C C C C5"/ I 5"/ I 5"/ 5"/ 5"/ I <5"/ 5"/! " ! " ! " '

!A* A

*< I< I A

5+ 5+ 5+ < I 5+ @ 5+ < I 5+ @@ @ *

!! ! ! ' '

'

'

'

"#$%&'(!=*!>5(!?;'(!2-!@)!

J(&' .75&' "2' @*' $,' )' %"+H&+$&+#' #""5' 2".' &,#$3)#$+/' #(&' 6"7G5$+/' #$3&' 2".' )+' $+H&,#3&+#' &).+$+/' $+#&.&,#'

%"34"7+6&6' )++7)5590' 8744",&' )' G)+K' 4)9,' LM' $+#&.&,#:' %"34"7+6&6' )++7)5590' N,&' #(&' .75&' "2' @*' #"'

)44."O$3)#&'#(&'6"7G5$+/'#$3&0'

+2';4,2/!

J(&'.75&'"2'@*',)9,'#()#'#(&'6"7G5$+/'#$3&'$,')44."O$3)#&59'@*

DL

'9&).,0'P2'9"7'7,&'#(&'2".375)'2".'%"34"7+6'

$+#&.&,#:'9"7'-$55',&&'#()#'#($,')44."O$3)#$"+'$,'&O%&55&+#0'

!!

! ! !"

!"#$%&'#()#'#(&'*')+,-&.,')/.&&0' '

!!

"#$%&'(!)*!+,%&',-.,/0!120$3,45%6!

1+&' "2' #(&' 3",#' $34".#)+#' 4."4&.#$&,' "2' 5"/).$#(3,' $,' #()#' #(&' 5"/).$#(3,' "2' 4."67%#,' ).&' #(&' ,73,' "2' #(&'

5"/).$#(3,0'8$3$5).59:'#(&'5"/).$#(3,'"2';7"#$&+#,').&'#(&'6$22&.&+%&,'"2'#(&'5"/).$#(3,0'

! 5+ * 5+< 5+ =*>=<> 5+? '

!*

5+ 5+ * 5+ *@*@

'

! *

AB AB AB5"/ CD 5"/ @ *5"/ @ '

!!

"#$%&'(!7*!83,4,/0!$!932:;<4!$6!$!+;%!

E"/).$#(3,'45)9')'3)F".'."5&'$+'%)5%757,'G9'%"+H&.#$+/'4."67%#,'#"',73,0!

! < <

C C C C C C C5"/ I 5"/ I 5"/ 5"/ 5"/ I <5"/ 5"/! " ! " ! " '

!A* A

*< I< I A

5+ 5+ 5+ < I 5+ @ 5+ < I 5+ @@ @ *

!! ! ! ' '

'

'

'

"#$%&'(!=*!>5(!?;'(!2-!@)!

J(&' .75&' "2' @*' $,' )' %"+H&+$&+#' #""5' 2".' &,#$3)#$+/' #(&' 6"7G5$+/' #$3&' 2".' )+' $+H&,#3&+#' &).+$+/' $+#&.&,#'

%"34"7+6&6' )++7)5590' 8744",&' )' G)+K' 4)9,' LM' $+#&.&,#:' %"34"7+6&6' )++7)5590' N,&' #(&' .75&' "2' @*' #"'

)44."O$3)#&'#(&'6"7G5$+/'#$3&0'

+2';4,2/!

J(&'.75&'"2'@*',)9,'#()#'#(&'6"7G5$+/'#$3&'$,')44."O$3)#&59'@*

DL

'9&).,0'P2'9"7'7,&'#(&'2".375)'2".'%"34"7+6'

$+#&.&,#:'9"7'-$55',&&'#()#'#($,')44."O$3)#$"+'$,'&O%&55&+#0'

!!

! !

!"

!"#$$%&!'(!)*%+#*,-#$!%.!"%/0*-,12$

!"#$%&'()*!!

! "#$%&$#'()*!+,-,)#$$.!/#0,!1!2&''(-*!3()!$(+#)4'/5*6!2#*,!78!9%(55(-!$(+#)4'/5:!#-;!2#*,! ! !9-#'&)#$!$(+#)4'/5:<!=*,!'/,!%/#-+,!(3!2#*,!3()5&$#!3()!('/,)!2#*,*<!

! >/,!)&$,!(3!?1!4*!3()!%(5@(&-;!4-',),*'A!#*!%(-')#*',;!B4'/!'/,!)&$,!(3!?8!3()!%(-'4-&(&*!%(5@(&-;4-+<!!

&

+(",-..*&

! >/,),! 4*! -(! 3()5&$#! 3()! '/,! $(+#)4'/5! (3! #! *&5! ()! ;433,),-%,<! C-! @#)'4%&$#)A!$(+ $(+ $(+ <" " "# $ # $ !

! D,! %#),3&$! (3! ;(5#4-! %/#-+,*! B/,-! &*4-+! @)(@,)'4,*! (3! $(+#)4'/5*<! E()! ,F#5@$,A! '/,! ;(5#4-! (31$-% & !4*!#$$! 8A& B/,),#*!'/,!;(5#4-!(3! 1$-% & !4*!#$$! 8<& !

! >/,!-('#'4(-!3()!$(+#)4'/5*!%#-!2,!%(-3&*4-+<!G(',!'/#'! $- $- <'& ' & !

&

!

/0! H0#$&#',!'/,!3($$(B4-+!$(+#)4'/5*!&*4-+!'/,!%/#-+,!(3!2#*,!3()5&$#<!I(&-;!.(&)!#-*B,)!'(!J!;,%45#$!@$#%,*<!

-0! J$(+ ? !!

10! 7K$(+ 7LM8 !!

20! I,B)4',!'/,!3($$(B4-+!,F@),**4(-*!4-!',)5*!(3! $- L !#-;! $-K< !

-0! $- 18 !

10! K$-ML!

30! =*,!'/,!@)(@,)'4,*!(3!$(+#)4'/5*!'(!),B)4',!#-;!*45@$43.!'/,!3($$(B4-+!$(+#)4'/54%!,F@),**4(-*!

-0! L$(+ N< !

10! M$- K <! !

40! =*,!'/,!@)(@,)'4,*!(3!$(+#)4'/5*!'(!,F@#-;!'/,!,F@),**4(-! $- ( !#*!#!*&5A!;433,),-%,A!#-;O()!%(-*'#-'!5&$'4@$,!(3!$(+#)4'/5*<!!

!!!!"#$%&'()

Topic 1 - Algebra (Exponents and Logarithms) 2

Page 3: 0)1) !#$%&'%()#*)+#,-'&./() +%((#0)1) !#$%&'%(dpmathematicssl.weebly.com/uploads/1/2/...algebra_laws_of_expone… · !"#$%&'#()#'#(&'*')+,-&.,')/.&&0' '!! "#$%&'(!)*!+,%&',-.,/0!120$3,45%6!

!"

!"#$$%&!'(!)*%+#*,-#$!%.!"%/0*-,12$

!"#$%&'()*!!

! "#$%&$#'()*!+,-,)#$$.!/#0,!1!2&''(-*!3()!$(+#)4'/5*6!2#*,!78!9%(55(-!$(+#)4'/5:!#-;!2#*,! ! !9-#'&)#$!$(+#)4'/5:<!=*,!'/,!%/#-+,!(3!2#*,!3()5&$#!3()!('/,)!2#*,*<!

! >/,!)&$,!(3!?1!4*!3()!%(5@(&-;!4-',),*'A!#*!%(-')#*',;!B4'/!'/,!)&$,!(3!?8!3()!%(-'4-&(&*!%(5@(&-;4-+<!!

&

+(",-..*&

! >/,),! 4*! -(! 3()5&$#! 3()! '/,! $(+#)4'/5! (3! #! *&5! ()! ;433,),-%,<! C-! @#)'4%&$#)A!$(+ $(+ $(+ <" " "# $ # $ !

! D,! %#),3&$! (3! ;(5#4-! %/#-+,*! B/,-! &*4-+! @)(@,)'4,*! (3! $(+#)4'/5*<! E()! ,F#5@$,A! '/,! ;(5#4-! (31$-% & !4*!#$$! 8A& B/,),#*!'/,!;(5#4-!(3! 1$-% & !4*!#$$! 8<& !

! >/,!-('#'4(-!3()!$(+#)4'/5*!%#-!2,!%(-3&*4-+<!G(',!'/#'! $- $- <'& ' & !

&

!

/0! H0#$&#',!'/,!3($$(B4-+!$(+#)4'/5*!&*4-+!'/,!%/#-+,!(3!2#*,!3()5&$#<!I(&-;!.(&)!#-*B,)!'(!J!;,%45#$!@$#%,*<!

-0! J$(+ ? !!

10! 7K$(+ 7LM8 !!

20! I,B)4',!'/,!3($$(B4-+!,F@),**4(-*!4-!',)5*!(3! $- L !#-;! $-K< !

-0! $- 18 !

10! K$-ML!

30! =*,!'/,!@)(@,)'4,*!(3!$(+#)4'/5*!'(!),B)4',!#-;!*45@$43.!'/,!3($$(B4-+!$(+#)4'/54%!,F@),**4(-*!

-0! L$(+ N< !

10! M$- K <! !

40! =*,!'/,!@)(@,)'4,*!(3!$(+#)4'/5*!'(!,F@#-;!'/,!,F@),**4(-! $- ( !#*!#!*&5A!;433,),-%,A!#-;O()!%(-*'#-'!5&$'4@$,!(3!$(+#)4'/5*<!!

!!!!"#$%&'()

!"

!"! !"#$%#&%'()%'%*+,%&&-"#' .# /.#0 12! ! '("'()%'."34,-()5'"6'4'&-#3.%'784#(-(9:'

#"! ;-#$'()%'%*4<('=4.8%'"6' /."3 > '?-()"8('8&-#3'4'<4.<8.4(",:'

$"! ;-#$'()%'%*4<('=4.8%'"6' / @.# .#" " '?-()"8('8&-#3'4'<4.<8.4(",:'

'

'

' '

Exponential and Logarithmic Equations

!!

!"#$$%&!'()!*+,%&#&-./0!/&1!"%2/3.-45.6!*78/-.%&$

!"#$%&%'()*+)%,+-$.)/('01(2+!34)'($%5+-&55$%+67+

!

!"#$%&'

! "#$%&'()*!')+(,+')-!,(-%.'&/0*!%)1!234()2)&*5!

! 644.(3'0%&2!*(,$&'()*5!

! 644,'7%&'()*5!

'

()**+,-'

8)!&/'*!,2**()9!:2!*(,+2!%!+%.'2&;!(<!2#$%&'()*!')+(,+')-!,(-%.'&/0*!%)1!234()2)&*5!!

'

./+*#01'23'("04$56'+5'./#"5157$+0'.8)+7$"5'

=(,+2!&/2!2#$%&'()!>

? @ >5!" !

("0)7$"5'

A2!$*2!&/2!4.(42.&'2*!(<!,(-%.'&/0*!%)1!234()2)&*!&(!*(,+2!<(.! 5! !

>

>

>

>

? @ >

? B

B

?

B,) ,)

?

B> ,)

?

C B,) D5CC> ?

!

!

!

!

""

"

"

!

!

! '!

!

./+*#01'93'("04$56'+':"6+,$7;*$%'.8)+7$"5'

E($!*/($,1!F2!7%.2<$,!&(!7/27G!;($.!%)*:2.*!%<&2.!*(,+')-!%)!2#$%&'()5!8)!&/'*!23%04,29!;($!:',,!*22!&/%&!()2!(<!

&/2!H*(,$&'()*I!'*!)(&!+%,'15!J/2!%442%.%)72!(<!&/2*2!H23&.%)2($*I!*(,$&'()*!(77$.*!<.2#$2)&,;!:'&/!,(-%.'&/0'7!

2#$%&'()*!F27%$*2!&/2!1(0%')!(<!&/2!,(-%.'&/0'7!<$)7&'()!'*!.2*&.'7&21!&(!4(*'&'+2!.2%,!)$0F2.*5!!

!!

!"#$$%&!'()!*+,%&#&-./0!/&1!"%2/3.-45.6!*78/-.%&$

!"#$%&%'()*+)%,+-$.)/('01(2+!34)'($%5+-&55$%+67+

!

!"#$%&'

! "#$%&'()*!')+(,+')-!,(-%.'&/0*!%)1!234()2)&*5!

! 644.(3'0%&2!*(,$&'()*5!

! 644,'7%&'()*5!

'

()**+,-'

8)!&/'*!,2**()9!:2!*(,+2!%!+%.'2&;!(<!2#$%&'()*!')+(,+')-!,(-%.'&/0*!%)1!234()2)&*5!!

'

./+*#01'23'("04$56'+5'./#"5157$+0'.8)+7$"5'

=(,+2!&/2!2#$%&'()!>

? @ >5!" !

("0)7$"5'

A2!$*2!&/2!4.(42.&'2*!(<!,(-%.'&/0*!%)1!234()2)&*!&(!*(,+2!<(.! 5! !

>

>

>

>

? @ >

? B

B

?

B,) ,)

?

B> ,)

?

C B,) D5CC> ?

!

!

!

!

""

"

"

!

!

! '!

!

./+*#01'93'("04$56'+':"6+,$7;*$%'.8)+7$"5'

E($!*/($,1!F2!7%.2<$,!&(!7/27G!;($.!%)*:2.*!%<&2.!*(,+')-!%)!2#$%&'()5!8)!&/'*!23%04,29!;($!:',,!*22!&/%&!()2!(<!

&/2!H*(,$&'()*I!'*!)(&!+%,'15!J/2!%442%.%)72!(<!&/2*2!H23&.%)2($*I!*(,$&'()*!(77$.*!<.2#$2)&,;!:'&/!,(-%.'&/0'7!

2#$%&'()*!F27%$*2!&/2!1(0%')!(<!&/2!,(-%.'&/0'7!<$)7&'()!'*!.2*&.'7&21!&(!4(*'&'+2!.2%,!)$0F2.*5!!

Topic 1 - Algebra (Exponents and Logarithms) 3

Page 4: 0)1) !#$%&'%()#*)+#,-'&./() +%((#0)1) !#$%&'%(dpmathematicssl.weebly.com/uploads/1/2/...algebra_laws_of_expone… · !"#$%&'#()#'#(&'*')+,-&.,')/.&&0' '!! "#$%&'(!)*!+,%&',-.,/0!120$3,45%6!

!!

!"#$$%&!'()!*+,%&#&-./0!/&1!"%2/3.-45.6!*78/-.%&$

!"#$%&%'()*+)%,+-$.)/('01(2+!34)'($%5+-&55$%+67+

!

!"#$%&'

! "#$%&'()*!')+(,+')-!,(-%.'&/0*!%)1!234()2)&*5!

! 644.(3'0%&2!*(,$&'()*5!

! 644,'7%&'()*5!

'

()**+,-'

8)!&/'*!,2**()9!:2!*(,+2!%!+%.'2&;!(<!2#$%&'()*!')+(,+')-!,(-%.'&/0*!%)1!234()2)&*5!!

'

./+*#01'23'("04$56'+5'./#"5157$+0'.8)+7$"5'

=(,+2!&/2!2#$%&'()!>

? @ >5!" !

("0)7$"5'

A2!$*2!&/2!4.(42.&'2*!(<!,(-%.'&/0*!%)1!234()2)&*!&(!*(,+2!<(.! 5! !

>

>

>

>

? @ >

? B

B

?

B,) ,)

?

B> ,)

?

C B,) D5CC> ?

!

!

!

!

""

"

"

!

!

! '!

!

./+*#01'93'("04$56'+':"6+,$7;*$%'.8)+7$"5'

E($!*/($,1!F2!7%.2<$,!&(!7/27G!;($.!%)*:2.*!%<&2.!*(,+')-!%)!2#$%&'()5!8)!&/'*!23%04,29!;($!:',,!*22!&/%&!()2!(<!

&/2!H*(,$&'()*I!'*!)(&!+%,'15!J/2!%442%.%)72!(<!&/2*2!H23&.%)2($*I!*(,$&'()*!(77$.*!<.2#$2)&,;!:'&/!,(-%.'&/0'7!

2#$%&'()*!F27%$*2!&/2!1(0%')!(<!&/2!,(-%.'&/0'7!<$)7&'()!'*!.2*&.'7&21!&(!4(*'&'+2!.2%,!)$0F2.*5!!

!!

!"#$$%&!'()!*+,%&#&-./0!/&1!"%2/3.-45.6!*78/-.%&$

!"#$%&%'()*+)%,+-$.)/('01(2+!34)'($%5+-&55$%+67+

!

!"#$%&'

! "#$%&'()*!')+(,+')-!,(-%.'&/0*!%)1!234()2)&*5!

! 644.(3'0%&2!*(,$&'()*5!

! 644,'7%&'()*5!

'

()**+,-'

8)!&/'*!,2**()9!:2!*(,+2!%!+%.'2&;!(<!2#$%&'()*!')+(,+')-!,(-%.'&/0*!%)1!234()2)&*5!!

'

./+*#01'23'("04$56'+5'./#"5157$+0'.8)+7$"5'

=(,+2!&/2!2#$%&'()!>

? @ >5!" !

("0)7$"5'

A2!$*2!&/2!4.(42.&'2*!(<!,(-%.'&/0*!%)1!234()2)&*!&(!*(,+2!<(.! 5! !

>

>

>

>

? @ >

? B

B

?

B,) ,)

?

B> ,)

?

C B,) D5CC> ?

!

!

!

!

""

"

"

!

!

! '!

!

./+*#01'93'("04$56'+':"6+,$7;*$%'.8)+7$"5'

E($!*/($,1!F2!7%.2<$,!&(!7/27G!;($.!%)*:2.*!%<&2.!*(,+')-!%)!2#$%&'()5!8)!&/'*!23%04,29!;($!:',,!*22!&/%&!()2!(<!

&/2!H*(,$&'()*I!'*!)(&!+%,'15!J/2!%442%.%)72!(<!&/2*2!H23&.%)2($*I!*(,$&'()*!(77$.*!<.2#$2)&,;!:'&/!,(-%.'&/0'7!

2#$%&'()*!F27%$*2!&/2!1(0%')!(<!&/2!,(-%.'&/0'7!<$)7&'()!'*!.2*&.'7&21!&(!4(*'&'+2!.2%,!)$0F2.*5!!

!"

!"#$%&'(%&%)*+',"-& #-. /0 #-./ 10 /#- 2! ! ! &

!"#$%&"'(

3"',4%&("5&5%&*6%&'(%&78"7%8',%6&"9&#":+8,'(;6&'"&6,;7#,9<&="'(&6,>%6&+->&8%;"$%&'(%&#":+8,'(;62&

/

/

/ /

/

#-. /0 #-./ 10 /#-

#- / / 1 #-

/ / 1

/ ? @

? @ A@ B A

! ! !! ! !

! ! !

! ! !! !! !

&

C(%8%& +8%& /& 6"#*',"-6& '"& '(,6& )*+>8+',4& %)*+',"-D& B! & +->& @2! &E"5%$%8F& B! ,6& +-& %G'8+-%"*6& 6"#*',"-&=%4+*6%&,'&,6&-"'&,-&'(%&>";+,-&"9&'(%&"8,:,-+#&%G78%66,"-2&C(%8%9"8%F&'(%&"-#<&6"#*',"-&,6& @2! &

&

)*+,-#.(/0(1--2"*&,+%&'3(%4.(!"#$%&"'("5(+'()6$+%&"'(

H-& '(,6& %G+;7#%F&5%& +8%& 9"84%>& '"& *6%& +& 4";7*'%8& "8& :8+7(,-:& 4+#4*#+'"8& '"& +778"G,;+'%& '(%& 6"#*',"-& "9& '(%&

%)*+',"-2&I778"G,;+'%&'(%&6"#*',"-&'"&'(%&%)*+',"-& /#- /2! ! & &&&

!"#$%&"'(

J8,'%&'(%&%)*+',"-&+6&+&9*-4',"-D& /. 0 #- /" ! ! ! 2&C(%&K%8"6&"9&'(,6&9*-4',"-&+8%&'(%&6"#*',"-6&'"&'(%&"8,:,-+#&%)*+',"-2&L6,-:&+&:8+7(,-:&*',#,'<F&<"*&6%%&'(+'&'(%&:8+7(&(+6&/&K%8"6F&+778"G,;+'%#<&A2B1M&+->&B2N@O2&3"'%&'(+'&,'&,6&,;7"66,=#%&'"&9,->&'(%&%G+4'&6"#*',"-6&'"&'(%&%)*+',"-2&

((

!%$78(9&-:&&

! C(%8%&;+-<&5+<6& '"&58,'%&+-&+-65%82&P"8& ,-6'+-4%F& ,-& '(%& 9,86'&%G+;7#%F&<"*&4"*#>&(+$%&58,''%-& '(%&

+-65%8&+6&B/B N N N N B#- F & #- F & #- F & #- F &"8& #-N #- O

/ O O O / /2&I##&"9&'(%6%&+8%&4"88%4'2(

! Q%4+##& '(+'& '(%& #":+8,'(;& "9& +& 6*;& ,6& -"'& '(%& 6*;& "9& '(%& #":+8,'(;62& C(+'& ,6F& '(%& %)*+',"-&

BA BA BA#": BA #": #": BA! ! & 9"8&+##&$+#*%6&"9& 2! &E"5%$%8F&<"*&4+-&6("5&'(+'& '(,6&%)*+',"-&>"%6&

(+$%&+&6"#*',"-D& BAR! 2(

(

( (

!"

!"#$%&'(%&%)*+',"-& #-. /0 #-./ 10 /#- 2! ! ! &

!"#$%&"'(

3"',4%&("5&5%&*6%&'(%&78"7%8',%6&"9&#":+8,'(;6&'"&6,;7#,9<&="'(&6,>%6&+->&8%;"$%&'(%&#":+8,'(;62&

/

/

/ /

/

#-. /0 #-./ 10 /#-

#- / / 1 #-

/ / 1

/ ? @

? @ A@ B A

! ! !! ! !

! ! !

! ! !! !! !

&

C(%8%& +8%& /& 6"#*',"-6& '"& '(,6& )*+>8+',4& %)*+',"-D& B! & +->& @2! &E"5%$%8F& B! ,6& +-& %G'8+-%"*6& 6"#*',"-&=%4+*6%&,'&,6&-"'&,-&'(%&>";+,-&"9&'(%&"8,:,-+#&%G78%66,"-2&C(%8%9"8%F&'(%&"-#<&6"#*',"-&,6& @2! &

&

)*+,-#.(/0(1--2"*&,+%&'3(%4.(!"#$%&"'("5(+'()6$+%&"'(

H-& '(,6& %G+;7#%F&5%& +8%& 9"84%>& '"& *6%& +& 4";7*'%8& "8& :8+7(,-:& 4+#4*#+'"8& '"& +778"G,;+'%& '(%& 6"#*',"-& "9& '(%&

%)*+',"-2&I778"G,;+'%&'(%&6"#*',"-&'"&'(%&%)*+',"-& /#- /2! ! & &&&

!"#$%&"'(

J8,'%&'(%&%)*+',"-&+6&+&9*-4',"-D& /. 0 #- /" ! ! ! 2&C(%&K%8"6&"9&'(,6&9*-4',"-&+8%&'(%&6"#*',"-6&'"&'(%&"8,:,-+#&%)*+',"-2&L6,-:&+&:8+7(,-:&*',#,'<F&<"*&6%%&'(+'&'(%&:8+7(&(+6&/&K%8"6F&+778"G,;+'%#<&A2B1M&+->&B2N@O2&3"'%&'(+'&,'&,6&,;7"66,=#%&'"&9,->&'(%&%G+4'&6"#*',"-6&'"&'(%&%)*+',"-2&

((

!%$78(9&-:&&

! C(%8%&;+-<&5+<6& '"&58,'%&+-&+-65%82&P"8& ,-6'+-4%F& ,-& '(%& 9,86'&%G+;7#%F&<"*&4"*#>&(+$%&58,''%-& '(%&

+-65%8&+6&B/B N N N N B#- F & #- F & #- F & #- F &"8& #-N #- O

/ O O O / /2&I##&"9&'(%6%&+8%&4"88%4'2(

! Q%4+##& '(+'& '(%& #":+8,'(;& "9& +& 6*;& ,6& -"'& '(%& 6*;& "9& '(%& #":+8,'(;62& C(+'& ,6F& '(%& %)*+',"-&

BA BA BA#": BA #": #": BA! ! & 9"8&+##&$+#*%6&"9& 2! &E"5%$%8F&<"*&4+-&6("5&'(+'& '(,6&%)*+',"-&>"%6&

(+$%&+&6"#*',"-D& BAR! 2(

(

( (

!"

!"#$%&'(%&%)*+',"-& #-. /0 #-./ 10 /#- 2! ! ! &

!"#$%&"'(

3"',4%&("5&5%&*6%&'(%&78"7%8',%6&"9&#":+8,'(;6&'"&6,;7#,9<&="'(&6,>%6&+->&8%;"$%&'(%&#":+8,'(;62&

/

/

/ /

/

#-. /0 #-./ 10 /#-

#- / / 1 #-

/ / 1

/ ? @

? @ A@ B A

! ! !! ! !

! ! !

! ! !! !! !

&

C(%8%& +8%& /& 6"#*',"-6& '"& '(,6& )*+>8+',4& %)*+',"-D& B! & +->& @2! &E"5%$%8F& B! ,6& +-& %G'8+-%"*6& 6"#*',"-&=%4+*6%&,'&,6&-"'&,-&'(%&>";+,-&"9&'(%&"8,:,-+#&%G78%66,"-2&C(%8%9"8%F&'(%&"-#<&6"#*',"-&,6& @2! &

&

)*+,-#.(/0(1--2"*&,+%&'3(%4.(!"#$%&"'("5(+'()6$+%&"'(

H-& '(,6& %G+;7#%F&5%& +8%& 9"84%>& '"& *6%& +& 4";7*'%8& "8& :8+7(,-:& 4+#4*#+'"8& '"& +778"G,;+'%& '(%& 6"#*',"-& "9& '(%&

%)*+',"-2&I778"G,;+'%&'(%&6"#*',"-&'"&'(%&%)*+',"-& /#- /2! ! & &&&

!"#$%&"'(

J8,'%&'(%&%)*+',"-&+6&+&9*-4',"-D& /. 0 #- /" ! ! ! 2&C(%&K%8"6&"9&'(,6&9*-4',"-&+8%&'(%&6"#*',"-6&'"&'(%&"8,:,-+#&%)*+',"-2&L6,-:&+&:8+7(,-:&*',#,'<F&<"*&6%%&'(+'&'(%&:8+7(&(+6&/&K%8"6F&+778"G,;+'%#<&A2B1M&+->&B2N@O2&3"'%&'(+'&,'&,6&,;7"66,=#%&'"&9,->&'(%&%G+4'&6"#*',"-6&'"&'(%&%)*+',"-2&

((

!%$78(9&-:&&

! C(%8%&;+-<&5+<6& '"&58,'%&+-&+-65%82&P"8& ,-6'+-4%F& ,-& '(%& 9,86'&%G+;7#%F&<"*&4"*#>&(+$%&58,''%-& '(%&

+-65%8&+6&B/B N N N N B#- F & #- F & #- F & #- F &"8& #-N #- O

/ O O O / /2&I##&"9&'(%6%&+8%&4"88%4'2(

! Q%4+##& '(+'& '(%& #":+8,'(;& "9& +& 6*;& ,6& -"'& '(%& 6*;& "9& '(%& #":+8,'(;62& C(+'& ,6F& '(%& %)*+',"-&

BA BA BA#": BA #": #": BA! ! & 9"8&+##&$+#*%6&"9& 2! &E"5%$%8F&<"*&4+-&6("5&'(+'& '(,6&%)*+',"-&>"%6&

(+$%&+&6"#*',"-D& BAR! 2(

(

( (

Topic 1 - Algebra (Exponents and Logarithms) 4

Page 5: 0)1) !#$%&'%()#*)+#,-'&./() +%((#0)1) !#$%&'%(dpmathematicssl.weebly.com/uploads/1/2/...algebra_laws_of_expone… · !"#$%&'#()#'#(&'*')+,-&.,')/.&&0' '!! "#$%&'(!)*!+,%&',-.,/0!120$3,45%6!

!"

!"#$%&'(%&%)*+',"-& #-. /0 #-./ 10 /#- 2! ! ! &

!"#$%&"'(

3"',4%&("5&5%&*6%&'(%&78"7%8',%6&"9&#":+8,'(;6&'"&6,;7#,9<&="'(&6,>%6&+->&8%;"$%&'(%&#":+8,'(;62&

/

/

/ /

/

#-. /0 #-./ 10 /#-

#- / / 1 #-

/ / 1

/ ? @

? @ A@ B A

! ! !! ! !

! ! !

! ! !! !! !

&

C(%8%& +8%& /& 6"#*',"-6& '"& '(,6& )*+>8+',4& %)*+',"-D& B! & +->& @2! &E"5%$%8F& B! ,6& +-& %G'8+-%"*6& 6"#*',"-&=%4+*6%&,'&,6&-"'&,-&'(%&>";+,-&"9&'(%&"8,:,-+#&%G78%66,"-2&C(%8%9"8%F&'(%&"-#<&6"#*',"-&,6& @2! &

&

)*+,-#.(/0(1--2"*&,+%&'3(%4.(!"#$%&"'("5(+'()6$+%&"'(

H-& '(,6& %G+;7#%F&5%& +8%& 9"84%>& '"& *6%& +& 4";7*'%8& "8& :8+7(,-:& 4+#4*#+'"8& '"& +778"G,;+'%& '(%& 6"#*',"-& "9& '(%&

%)*+',"-2&I778"G,;+'%&'(%&6"#*',"-&'"&'(%&%)*+',"-& /#- /2! ! & &&&

!"#$%&"'(

J8,'%&'(%&%)*+',"-&+6&+&9*-4',"-D& /. 0 #- /" ! ! ! 2&C(%&K%8"6&"9&'(,6&9*-4',"-&+8%&'(%&6"#*',"-6&'"&'(%&"8,:,-+#&%)*+',"-2&L6,-:&+&:8+7(,-:&*',#,'<F&<"*&6%%&'(+'&'(%&:8+7(&(+6&/&K%8"6F&+778"G,;+'%#<&A2B1M&+->&B2N@O2&3"'%&'(+'&,'&,6&,;7"66,=#%&'"&9,->&'(%&%G+4'&6"#*',"-6&'"&'(%&%)*+',"-2&

((

!%$78(9&-:&&

! C(%8%&;+-<&5+<6& '"&58,'%&+-&+-65%82&P"8& ,-6'+-4%F& ,-& '(%& 9,86'&%G+;7#%F&<"*&4"*#>&(+$%&58,''%-& '(%&

+-65%8&+6&B/B N N N N B#- F & #- F & #- F & #- F &"8& #-N #- O

/ O O O / /2&I##&"9&'(%6%&+8%&4"88%4'2(

! Q%4+##& '(+'& '(%& #":+8,'(;& "9& +& 6*;& ,6& -"'& '(%& 6*;& "9& '(%& #":+8,'(;62& C(+'& ,6F& '(%& %)*+',"-&

BA BA BA#": BA #": #": BA! ! & 9"8&+##&$+#*%6&"9& 2! &E"5%$%8F&<"*&4+-&6("5&'(+'& '(,6&%)*+',"-&>"%6&

(+$%&+&6"#*',"-D& BAR! 2(

(

( (

!"

!!!"#$$%&!'()!*+,%&#&-./0!/&1!"%2/3.-45.6!*78/-.%&$

!"#$%&&'(! !"#$%&'()*"+,"%")%-&$"&,./&$0%12"322,%24+5"%+,-%+((*)4&'+"2%+,$32*$6%7)*%&,$"+,528%"02%24+5"%+,$32*%")%

"02%29/+"&),% : ; <, =! %&$%>; > 6! "

"%?0&$%&$%+((*)4&'+"2<@%29/+<%")%A6B>8%5)**25"%")%;%-25&'+<%

(<+52$6%%

! C2%5+*2D/<%)D%24"*+,2)/$%$)</"&),$%302,%$)<E&,.%29/+"&),$%&,E)<E&,.%<).+*&"0'$%+,-%24(),2,"$6%F+G2%$/*2%"0+"%@)/%5025G%@)/*%+,$32*$%&,%"02%)*&.&,+<%29/+"&),6%

(

%

)*! H)<E2%"02%D)<<)3&,.%24(),2,"&+<%29/+"&),$6%

%*! % = >B! %

+*! > B=I

!

%

,*! H)<E2%"02%D)<<)3&,.%<).+*&"0'&5%29/+"&),$6%

%*! <, J! %

+*! <,K; >L :! %

-*! H)<E2%"02%D)<<)3&,.%24(),2,"&+<%29/+"&),$6%M)/,-%@)/*%*2$/<"%")%N%-25&'+<%(<+52$6%

%*! NI NBA! %

+*!>;A6>A> ;

>;

#

%

.*! ; = : A! !" " %

/*!; ;! !" " %

0*! H)<E2%"02%D)<<)3&,.%<).+*&"0'&5%29/+"&),$6%M)/,-%@)/*%*2$/<"%")%N%-25&'+<%(<+52$6%

%*! : :<). N ; <). B! ! %%

+*! = =><). <). >;

! ! %%

!!!!"#$%&'()

Topic 1 - Algebra (Exponents and Logarithms) 5

Page 6: 0)1) !#$%&'%()#*)+#,-'&./() +%((#0)1) !#$%&'%(dpmathematicssl.weebly.com/uploads/1/2/...algebra_laws_of_expone… · !"#$%&'#()#'#(&'*')+,-&.,')/.&&0' '!! "#$%&'(!)*!+,%&',-.,/0!120$3,45%6!

!"

!!!"#$$%&!'()!*+,%&#&-./0!/&1!"%2/3.-45.6!*78/-.%&$

!"#$%&&'(! !"#$%&'()*"+,"%")%-&$"&,./&$0%12"322,%24+5"%+,-%+((*)4&'+"2%+,$32*$6%7)*%&,$"+,528%"02%24+5"%+,$32*%")%

"02%29/+"&),% : ; <, =! %&$%>; > 6! "

"%?0&$%&$%+((*)4&'+"2<@%29/+<%")%A6B>8%5)**25"%")%;%-25&'+<%

(<+52$6%%

! C2%5+*2D/<%)D%24"*+,2)/$%$)</"&),$%302,%$)<E&,.%29/+"&),$%&,E)<E&,.%<).+*&"0'$%+,-%24(),2,"$6%F+G2%$/*2%"0+"%@)/%5025G%@)/*%+,$32*$%&,%"02%)*&.&,+<%29/+"&),6%

(

%

)*! H)<E2%"02%D)<<)3&,.%24(),2,"&+<%29/+"&),$6%

%*! % = >B! %

+*! > B=I

!

%

,*! H)<E2%"02%D)<<)3&,.%<).+*&"0'&5%29/+"&),$6%

%*! <, J! %

+*! <,K; >L :! %

-*! H)<E2%"02%D)<<)3&,.%24(),2,"&+<%29/+"&),$6%M)/,-%@)/*%*2$/<"%")%N%-25&'+<%(<+52$6%

%*! NI NBA! %

+*!>;A6>A> ;

>;

#

%

.*! ; = : A! !" " %

/*!; ;! !" " %

0*! H)<E2%"02%D)<<)3&,.%<).+*&"0'&5%29/+"&),$6%M)/,-%@)/*%*2$/<"%")%N%-25&'+<%(<+52$6%

%*! : :<). N ; <). B! ! %%

+*! = =><). <). >;

! ! %%

!!!!"#$%&'()

!"

!"! !"#$ %$ &'%()*+&$ ,-*.*-/$ -0$ %(('01*2%-#$ -)#$ "0.,-*0+$ 03$ -)#$ #4,%-*0+$ 567.0& 58! ! $ %99,'%-#$ -0$ 5$$

:#9*2%.$(.%9#";$

$

$ $

Topic 1 - Algebra (Exponents and Logarithms) 6

Page 7: 0)1) !#$%&'%()#*)+#,-'&./() +%((#0)1) !#$%&'%(dpmathematicssl.weebly.com/uploads/1/2/...algebra_laws_of_expone… · !"#$%&'#()#'#(&'*')+,-&.,')/.&&0' '!! "#$%&'(!)*!+,%&',-.,/0!120$3,45%6!

Answers:

Properties of Logarithms

!"#

!"! "#$%&'(! )*! !+,-.&/%0!%12$3.#.,(! )*! !&'.,-/,3.(! 45 !4 6 !

!!!! !!

#"! "#$%&'(! 4*! !+,-.&/%0!%12$3.#.,(! 4*! !&'.,-/,3.(! )5 !4 6 !!

!!!!!! !

$%&&'()*)

+"! ,"! 70' 8 )69:;90#< 8 )688)0'7 )649=>

!

!

!"!);

0'):>4 86?=>?0#< ):>4 ?6>9)0'); ?684=) !

!6!

-"! ,") 0' ?4 0'@: ;A 0' : 0'; !

!")7;0' 0'; 0' >: 0'; 0' : 0' ; 70' :

>:!

."! ,"!)7 ?

: : : : :) 70#< = 0' ? 70' ? 70' : 7 0' :? ?

!

!") > >0' ; 0' ; 0' 0' ; > 0' 0' ; >" " " !

!"#

!"#$%&"'(

!"!!" !#$ #$ #$

"! ! ! %

#"! &&#$ &#$' !( #$ #$ ! #$

!"" " " "

"%

$"! "& & &#)* + #)* & "#)* & "'!( " %

%"! & ,#$ #$ &#$ , #$ & , -# # # # %%

&'(()*+,-%

,"! ."% "- !. - "" " %

/"+"! .- / / " "

/

"" " "

+0"! ."% ,#$ ," " # %

/"+0 0 0!#$'" !( 0 " ! " ! !

"" " # " # " #

+

1"! ."% & & #$ &.1/ &.1 #$/ & #$/ #$ &.1 12+--&#$/

" " " " %

%

!"!

/"++

!"12!1! "!"

$

%

%%%%%%%%%%%%%%%%%%%%%%%

!"!"2! "!"!"2!!" #$ #$ "!"

! #$ " .2+.1!"2!!" #$ !"

$

$

$

%

%2"% " - 0 1" "# # %

%%%%%%%%

0 #$ 0 !2.1+

!3 456)7748#92

"

"

# "#

%

%

3"%%" " " "" "# # " " %

%%

" " 1" ! 1"3 % !

" "" "" %

Exponential and Logarithmic Equations

!"#

!"#$%&"'(

!"!!" !#$ #$ #$

"! ! ! %

#"! &&#$ &#$' !( #$ #$ ! #$

!"" " " "

"%

$"! "& & &#)* + #)* & "#)* & "'!( " %

%"! & ,#$ #$ &#$ , #$ & , -# # # # %%

&'(()*+,-%

,"! ."% "- !. - "" " %

/"+"! .- / / " "

/

"" " "

+0"! ."% ,#$ ," " # %

/"+0 0 0!#$'" !( 0 " ! " ! !

"" " # " # " #

+

1"! ."% & & #$ &.1/ &.1 #$/ & #$/ #$ &.1 12+--&#$/

" " " " %

%

!"!

/"++

!"12!1! "!"

$

%

%%%%%%%%%%%%%%%%%%%%%%%

!"!"2! "!"!"2!!" #$ #$ "!"

! #$ " .2+.1!"2!!" #$ !"

$

$

$

%

%2"% " - 0 1" "# # %

%%%%%%%%

0 #$ 0 !2.1+

!3 456)7748#92

"

"

# "#

%

%

3"%%" " " "" "# # " " %

%%

" " 1" ! 1"3 % !

" "" "" %

Topic 1 - Algebra (Exponents and Logarithms) 7

Page 8: 0)1) !#$%&'%()#*)+#,-'&./() +%((#0)1) !#$%&'%(dpmathematicssl.weebly.com/uploads/1/2/...algebra_laws_of_expone… · !"#$%&'#()#'#(&'*')+,-&.,')/.&&0' '!! "#$%&'(!)*!+,%&',-.,/0!120$3,45%6!

!"#

!"#$%&"'(

!"!!" !#$ #$ #$

"! ! ! %

#"! &&#$ &#$' !( #$ #$ ! #$

!"" " " "

"%

$"! "& & &#)* + #)* & "#)* & "'!( " %

%"! & ,#$ #$ &#$ , #$ & , -# # # # %%

&'(()*+,-%

,"! ."% "- !. - "" " %

/"+"! .- / / " "

/

"" " "

+0"! ."% ,#$ ," " # %

/"+0 0 0!#$'" !( 0 " ! " ! !

"" " # " # " #

+

1"! ."% & & #$ &.1/ &.1 #$/ & #$/ #$ &.1 12+--&#$/

" " " " %

%

!"!

/"++

!"12!1! "!"

$

%

%%%%%%%%%%%%%%%%%%%%%%%

!"!"2! "!"!"2!!" #$ #$ "!"

! #$ " .2+.1!"2!!" #$ !"

$

$

$

%

%2"% " - 0 1" "# # %

%%%%%%%%

0 #$ 0 !2.1+

!3 456)7748#92

"

"

# "#

%

%

3"%%" " " "" "# # " " %

%%

" " 1" ! 1"3 % !

" "" "" %

!""

!!"! #"! " "#$% & ' #$% (! ! !

& ' () )*

! !!!

!$"! ) )

*#$% #$% *'

! !

!

!!!!!!!!!!!!!!

)

*'

*#$%* '

) '*'+ *, ' ''

!!

!!! ! !!

!

!

%"! -./0%!1!%2134/0%!56/#/678!%2134!649!:50;6/$0! &*<#$% &=" ! ! !>$5!?/##!$@61/0!649!'!A92$.8!

*=)(B 10C <=<<*=! ! !

&

'())*+&,,&

,"! D9!?106!6$!:/0C!649!6/E9!099C9C!:$2!649!1E$506!6$!291;4!F'<8<<<=!

<=<&"

<=<&"

'<8<<< *<8<<<

'#0 ' <=<&"

#0 ' *B=G 7912.<=<&"

#

#

$$

#

#

!

-"! H96!% !@9!649!/0/6/1#!1E$506=!D9!:/2.6!5.9!649!41#:I#/:9!/0:$2E16/$0!6$!;1#;5#169!649!J1#59!$:! =& !

!

+*"BB, *"BB* *' '

#0 * '*#0 *"BB' *"BB

& &% %$ $

& &!

K$?!?9!;10!5.9!649!C9;17!E$C9#!6$!:/0C!649!1E$506!1:692!*<<<!7912.=!#0 * ' L*"BB *<<<*< (=)G %21E.&#" %$ $ !

Topic 1 - Algebra (Exponents and Logarithms) 8