Transcript LOGARITHMS

Logarithms
Laws of logarithms
Product Rule
loga xy = loga x + loga y
loga x + loga y = loga xy
Examples:
log5 (35) = log5 3 + log5 5
log3 5 + log3 4 = log3 (5  4)
= log3 20
Quotient Rule
x
loga ( ) = loga x – loga y
y
x
loga x – loga y = loga ( )
y
log5 20 – log5 4 = log5 ( 20 )
4
= log5 5
=1
16
log3( ) = log3 16 – log3 5
5
Power Rule
loga xm = m loga x
m loga x = loga xm
logx 53 = 3 logx 5
4 log9 3 = log9 34
= log9 81
= log9 92
= 2 log9 9
=2
(log9 9 = 1)
Express the following as a single logarithms
loga 3 + loga 4 – loga5
loga 3 + loga 4 – loga5
= loga (3  4) – loga 5
= loga12 – loga 5
 12 
= loga  5 
 
= loga 2.4
(Rule 1)
(Rule 2)
Express the following as a single logarithms
5 log4 x – 2 log4 y + 3 log4 z
= log4 x5 – log4 y2 + log4 z3 (Rule 3)



4 

 log x
y

5

2 

+ log4 z3
 x 5z 3 
= log4  2 
 y 


(Rule 2)
(Rule 1)
Question:
Given that log2 3 = 1.58
and log2 5 = 2.32,
Find value of each of the following.
(a) log2 75
(b) log2 0.3
(c) log2 √5
SOLUTION
Given that log2 3 = 1.58 and log2 5 = 2.32
(a) log2 75 = log2 [325]
= log2 3 + log2 25
= log2 3 + log2 52
= log2 3 + 2 log2 5
= 1.58 + 2(2.32)
= 6.22
Rule 1
Solution
(b) log2 0.3 = log2 (3÷10)
= log2 3  log2 10
Rule 2
= log2 3  [log2 (52)]
= log2 3  [log2 5 + log2 2]
= 1.58  (2.32 + 1)
=  1.74
(c) log2 √5 = (1/2) log2 5
= (1/2)(2.32)
= 1.16
CHANGE OF BASE
Change of base-a to base c is as follows:
loga b = logc b
logc a
For example, to change log4 8 to base-2
log4 8 = log2 8
log2 4
log2 23
3


log2 22
2
EXAMPLE
Evaluate log5 12.
log1012
log5 12 = log 5
10
1
.
0792

0.6990
 1.544
Use calculator
Use at least 4 significant figures
CHANGE OF BASE
Change of base-a to base b is as follows:
log
b
b
loga b =
logb a
1

logb a
For example, to change log32 2 to base-2
1
log32 2 =
log2 32
1
1


5
5
log2 2
EXERCISE
Given that log2 5 = 2.32 find the value
for each of the following without using
calculator. ( without changing to base-10)
(a) log5 4
(b) log5 2
(c) log4 50
Given that log2 5 = 2.32
(a) log5 4  log2 4
log2 5
log2 2 2

log2 5

2
log2 5
 2
2.32
=0.8621
Change to base-2
Rule 3
1
1 =0.431
(b) log5 2 =

log2 5 2.32
log
50
2
(c) log4 50 
log2 4
log2( 2  25)

log2 22
log2 2  log2 52

2
1  2 log2 5

2
 1  log2 5
2
 0.5  2.32
=2.82