Logarithms

LOGARITHMS WHEN WE ARE GIVEN the base 2, for example, an exponent !, then "e #an e$al%ate 2 !& 2! ' (& In$ersel), *f "e are +*$en the base 2 an *ts po"er (  2? ' (  then "hat *s the exponent exponent that  that "*ll pro%#e ( That exponent exponent *s #alle a lo+ar*thm lo+ar*thm&& We #all the exponent exponent ! the  thelo+ar*thm lo+ar*thm of ( "*th base 2& 2 & We "r*te ! ' lo+2(& We "r*te the base 2 as a s%bs#r*pt& ! *s the exponent  to  to "h*#h 2 m%st be ra*se to pro%#e (& A lo+ar*thm *s an exponent& S*n#e ./0 ' ./,/// then lo+././,/// ' 0& 1The lo+ar*thm of ./,/// "*th base ./ *s 0&1 0 *s the exponent  to  to "h*#h ./ m%st be ra*se to pro%#e ./,///& 1./0 ' ./,///1 *s #alle the exponent*al form& form& 1lo+././,/// ' 01 *s #alle the lo+ar*thm*# form& form& Here *s the en*t*on3 lo+b x   x  '  ' n  means bn '  ' x   x &  That base "*th that exponent exponent   pro%#es x  pro%#es x & Example 1. Wr*te *n exponent*al form3 Answer&& 24 ' !2& Answer

lo+ 2!2 ' 4&

. .6

Example 2. Wr*te *n lo+ar*thm*# form3 0 52 ' Answer.

 . .6

lo+0

&

 ' 52&

Problem 1. Wh*#h n%mbers ha$e ne+at*$e lo+ar*thmsTo see the answer, pass your mouse over the colored area. To cover the answer again, click "Refresh" ("Reload"). o the problem yourself !rst Proper fractions. Example 3. E$al%ate lo+(.& Answer& ( to "hat exponent pro%#es .- (/ ' .& lo+(. ' /& We #an obser$e that, *n an) base, the lo+ar*thm of . *s /& lo+b. ' / Example 4.

E$al%ate lo+44&

Answer& 4 "*th "hat exponent "*ll pro%#e 4- 4 . ' 4& Therefore, lo+44 ' .& In an) base, the lo+ar*thm of the base *tself *s .& lo+bb ' . Example 5 . lo+22m ' Answer& 2 ra*se to "hat exponent "*ll pro%#e 2m - m, ob$*o%sl)& lo+22m ' m&  The follo"*n+ *s an *mportant formal r%le, $al* for an) base b3 lo+bb x  ' x   Th*s r%le embo*es the $er) mean*n+ of a lo+ar*thm&  x   on the r*+ht  *s the exponent  to "h*#h the base b m%st be ra*se to pro%#e b x &  The r%le also sho"s that the exponent*al f%n#t*on b x  *s the *n$erse of the f%n#t*on lo+b x & We "*ll see th*s *n the follo"*n+ Top*#& Example 6 . Answer. lo+!

. 7

E$al%ate lo+! .  *s e8%al to ! "*th "hat exponent7 . 7

'

9ompare the pre$*o%s r%le&

lo+!!52  ' 52&

.  ' !52& 7

&

Example 7. lo+2 .24 ' Answer& .24 ' : ' 2 52& Therefore, lo+2 .24 ' lo+2252 ' 52& Example 8. lo+!

 ' -

' !.;4& Therefore,

Answer.

lo+!

 ' lo+!!.;4 ' .;4&

Problem 2. Wr*te ea#h of the follo"*n+ *n lo+ar*thm*# form& To see the answer, pass your mouse over the colored area. To cover the answer again, click "Refresh" ("Reload"). a>

bn ' x 

logb x   n

b>

2! ' (

log28  3

#>

./2 ' .//

log1!1!!  2

>

452 ' .;24&

log51"25  #2.

Problem 3. Wr*te ea#h of the follo"*n+ *n exponent*al form& bn   x 

a> lo+b x  ' n

82  64

#> 2 ' lo+(60

b> lo+2!2 ' 4

25  32

> lo+6.;!6 ' 52

652 ' 652 ' .;!6 .;!6

Problem 4. E$al%ate the follo"*n+& 4

a> lo+2.6

3

#> lo+4.24

> lo+(.

1

e> lo+((

2

b> lo+0.6

!

f> lo+./.

!

Problem 5. What n%mber *s n1!!!

a> lo+./n ' !

#> lo+2n ' /

e> lo+n

1 . .6

 ' 52

4

b> 4 ' lo+2n

32

> . ' lo+./n

1!

f> lo+n

. 4

 ' 5.

5

+> lo+2

. !2

 ' n

. 2

h> lo+2

#5

 ' n

#1

Problem 6. lo+bb x  '  x  Problem 7. E$al%ate the follo"*n+& . 7

a> lo+7

lo+775. ' 5.

b> lo+7

 . (.

#2

#> lo+2

. 0

 #2

> lo+2

. (

#3

e> lo+2

 . .6

50

f> lo+./ &/.

 #2

+> lo+./ &//.

#3

h> lo+6

1"3

*> lo+b

 3"4

 9ommon lo+ar*thms  The s)stem of #ommon lo+ar*thms has ./ as *ts base& When the base *s not *n*#ate, lo+ .// ' 2 then the s)stem of #ommon lo+ar*thms  base ./  *s *mpl*e& Here are the po"ers of ./ an the*r lo+ar*thms3 Powers of 1!$

. .///

. .//

%ogarit&ms$

5!

52

. ./

5.

.

./

.//

.///

/

.

2

!

./,///

0

Lo+ar*thms repla#e a +eometr*# ser*es "*th an ar*thmet*# ser*es& Problem 8. lo+ ./n ' -

n. '&e base is 1!.

Problem (. lo+ 4( ' .&?6!0&

Therefore, ./.&?6!0  ' -

58. 1.7634 is t&e common logarit&m of 58. )&en 1! is raise* to t&at exponent+ 58 is pro*,ce*. Problem 1!. lo+ ' .& What n%mber *s x -

log a  1+ implies a  1!. -ee abo/e.0 '&erefore+ log -log  x 0  1 implies log  x    1!. ince 1! is t&e base+  x   1!1!  1!+!!!+!!!+!!! Nat%ral lo+ar*thms  The s)stem of nat%ral lo+ar*thms has the n%mber #alle e as *ts base&