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J . .
I d'�nderstanding the Value of a Logarithm
The expression�· s a logarithm and is read "�o� base b of a,:· where� and� Study the table)flogarithmic equations below. In each box, write an exponential equation that is related with the given logarithmic equation.
I
�3 27 = 31
33
= �'1
log 1o 1000 = 3(
,cl= \00()
}009 243 = 2e, 2
9�'/.� -::- .D.J..t � log4 4 V4 = f
i ""''�-= "°t � -=: 't • L\ \I�
log5 iis = -3
-3 \ 5 -= ,�s
log216=4'
�'-4 =\b
1 --og36 6 = 2
'3 f.o "� ::: "log 10 0.000001 = -6""
,o-.,= o.cOOOO\
. I log3 3s =l . 5
�\ls :: 3 ''slog 5 5 = 1
s' - 5
.
og51Z5 =3
5�= \.l.5 log27 9 = t
� 1.2/�-=- 9
log 5 .Js = t ,s'f!)..
::. JS
log4 _L = -216
-?. _L � = lb
1086 W =t
e,''3-=- �
Based on studying the examples2 make a conjecture as to how one would find the value of log b a .
� "ii4w_al, �..,..�::::
� . �v� � °'-. ��. \s· ���-#.cl-� ts f\i)J..� -fu � or-cl.or --\.o � ���
Make three conjectures about the value of a logarithm based on the numerical relationship between the value of the base (b) and the value of t!1e argument (a) of the logarithm..
If� = �' then ...
If b < a, then ...
f b > a, then ...
.
A logarithm whose base is IO is called a commoWea,·ithm and is written � = log 1 oo . Alogarithm with an unwlitten base is understood to have a base of 10. Use a calculator to find the values of .. each of the following common logarithms .
log 5 log 100 log 115
D .. bq q � �,Ob\
•
Based on your understanding of the value of a logarithm, what conclusion can you make about the values of the following logaritluns?
'
Logarithmic Expression .
log 3 81 ,:: )(.
log 2 t -::- )(
log 2 9 -:: X
log 3 36 -=- '/..
log 4 21 -=- )(.
log 3 105 -= "1-,
'
Find the value of the given logarithm. If the value is not an
integer, between what two integer values do you think the value of the
logarithmic expression lies?
3 X -::.8\
BJ 3)(
-::34
•
.l".,, t 5 g."' = � "'r1
:). V. - -cg-\ l 1'.:::. � -3 - . l-3:)...,. :: (q
� �?,
::; s �"=-\lo �
3)<-=-,3.b� �3�,, ;i; ?,\: :3I 3a...A..1 I
4)(
-=-r\�� '::: \{, 4 3 _ l, "\ \�)
.f � 3 X -:=\05
� 34
::�\ 3sf,Lt3 1 w..�
log a What can you conclude about the values of log b a and -1 b ?. . og
Given the expression log b a ,
use the calculator to find the log a
value of lb. . og
� - 1 -�3
-. �
.
.03c�) - -�-\Jl)Q ;;...
(\ ' -�sq 3. \ '10-�� .
���
�3 � 3��,�
�5�\ - ;) ,, \ 'tb
�l\
� 4 .. "J.?>b -
-
Le� '3 .
.\' ·
Developing the Natural Base, e
Consider the expressi@Complete the table below for the given values of n.
n
1
\\J�J 10.
100
1000
\ � HJ�OCV 10,000
\ ! Yi 100,000
'(( ,) '=.1,000,000
10,000,000
(1+�f "
�.sq�
�. '105 �.'1\'1 !l .. .,, �
�.r-tlP, �.'1\8
9'.'1lS
'1r' � 3.\4 t. '::? � .. rt ( '6
As n � oo, the_�/illl�i�
-=� � • '11 S . This value is called;',.
and is Called the ,nfthlral base.
� © r ,·
A logaritlun whose base is � is called a !].atural lQgaritlup. �s written as \in a\ All of.the prbpertiesof logaritluns that exist for bases other than e also exist for natural logaritluns. � 5 =- l • (o O't
Given the expression log b a , us_e the Given the expression log b a ,
use the calculator to find the Logarithmic Expression value of In a •
log 3 20.
.
���3
log2 1�
log 5 15
.i��
.�'-U �0$
�lt)
�?.-
�% \5
�5
log a 1 What can you conclude about the values of log b a and -1 b and .J!.Q. ?og 1n b
-
In b
..... � I '121
Find the value of x in each of the logarithmic equations below by rewriting the equation as an exponential equation.
log 3 (2x + 1) = -3
�-?>_ �x-t \
�)"'
-= ��+ \,_!_ - � -= :i...�9-1 ::i.1
- �\a ::: �)(;).1
log 2 9 = X + 2
••
lnx=3
log25 5.Js = X
a t> -,. = 51 • 5 'l?--
SJ°)( -= 5 �..,.
In(x+ 3) = 5
ec; -=-Xt3
�s
-3 -::.. X
G :: l 4 5. 4 I �J
Name A:n!?� K� Date --------Period ---
Logarithms Practice
Find the exact value of each of the following logaritlunic expressions without the aid of a calculator.
1. log2 32 :: )(.
� x :: 3:2
,_.,. : �a;
\!� ID 5. loglOO � )<.
\ox :: 10 D \ o
>< = IO i..
\x=2] -9. log2(4-8 2 ) :. X.
'2;i >< -: 4·1 )(. - i '2. 2'-:>. - •
�� -:: 2. !._ �X-= B \
2. log5 5 :- �
S X � ':, \
LX�D
6. log0.0001 = ;,<.
10 x = 0.000\ \ c
>< = 'o-'i
t'><: -1]
10. I
log6 6 · 62 :. "j....
G:, � = <a. b,,�
toy. :: fo 3/�
�x-:: 3/;\
3. log381 = )(. 4. log5125 : 'X.
3 ..,. = 8\ s)t=- ,1s
..... S)(= S� 3>'-
.:3
fx =-� \>< :.3) 7. log4 4-2 : X. 8. log2 2'ifi. -:.. �
4>' = 4' .. � \/3
2)( : � . .;2
�" : ';).. 4/3 �= -:iJ \x; 44/;1
11. log2 �) = X 12. log3 (s\-) = X.
,;2.�= ..L �
)( - ..l-3 - Bl
')...,.. : � ... 3-4
a"'" = 3
t>< = -fil t,�= -11 Given the lo arithmic ex ression a detem1ine between which two inte ers that value should lie without g p ,( ) g using a calculator, with reasoning, and (b) the value to three decimal places using a calculator, showing your work.
13. log35 (a) \ 4. �\t;� � b\c.. ?,� �"�J '�
be.��'=;� &2.=C\ 14. log2 21 (a) �'-lo�� '2.\ <. s
�C.. 2-\, ��v�°tJ {�
be.� ::). ":: \ l.o � � 5 :. 3:l. 15. log5156 (a) 3 � �':. \')(. � 4-
b\c. \ -S<o, -t\u. �ur,..c.4J \ ��� s'=,').s�s"' =�'-S
Cb) � - �sLii�� -w = v.•H.5\
(b) � 2. \ ..lM. 'l.\
� :i. = J,,...7- :�3�:;.\..�
(b)
�\�C.:: ��"�\3.13e} L,o�
Solve each of the following equations. Round your answers to three decimal places, if necessary.
16. log3 (x + 2) = 2
3 :2.=- x+.2..
� :: x+ '2.
� )( :: riJ
19. ln(2x + 3) = 3
e3 = 2..,c..+�
e.3 -3 : :>. )(.
�-= e..�-3
L )C. -=- i . s itfil
17.
20.
ln(x-3) = 2
e"= x-3 2.
)(.:: e. -t3
()(.: (0. 34&�
log2(3x) = -3
l-� = 3x...L- 3)(. t ?- = M
18. log9(x) =-1
9-' :: 'X.
\,\: -0
21. ln(x + 2) = -2
e ·2. -:: X °"" -:2..
')(.:. -1-1 e
\ -i"'= -e'l,.
�: -\. ii.s\
(_
1...--..
r'....___)