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'....___)