Transcript More Laws of Logarithms

“Teach A Level Maths”
Vol. 1: AS Core Modules
48: More Laws of
Logarithms
© Christine Crisp
The Laws of Logs
Module C2
"Certain images and/or photos on this presentation are the copyrighted property of JupiterImages and are being used with
permission under license. These images and/or photos may not be copied or downloaded without permission from JupiterImages"
The Laws of Logs
Log laws for Multiplying and Dividing
We’ll develop the laws by writing an
example with the numbers in index form.
The Laws of Logs
42  26  1092
A log is just an index, so to write this in index
form we need the logs from the calculator.
log 10 42  1 623 and log 10 26  1 415
1 623
1 415

42  26  10
 10
1 623  1  415
 10
 10
3  038
 log 10 (42  26)  3  038
So,
log 10 (42  26) 
(  1092 )
The Laws of Logs
42  26  1092
A log is just an index, so to write this in index
form we need the logs from the calculator.
log 10 42  1 623 and log 10 26  1 415
1 623
1 415

42  26  10
 10
1 623  1  415
 10
 10
3  038
(  1092 )
 log 10 (42  26)  3  038
So,
log 10 (42  26)  log 10 42 
The Laws of Logs
42  26  1092
A log is just an index, so to write this in index
form we need the logs from the calculator.
log 10 42  1 623 and log 10 26  1 415
1 623
1 415

42  26  10
 10
1 623  1  415
 10
 10
3  038
(  1092 )
 log 10 (42  26)  3  038
So,
log 10 (42  26)  log 10 42  log 10 26
The Laws of Logs
42  26  1092
A log is just an index, so to write this in index
form we need the logs from the calculator.
log 10 42  1 623 and log 10 26  1 415
1 623
1 415

42  26  10
 10
1 623  1  415
 10
 10
3  038
(  1092 )
 log 10 (42  26)  3  038
So,
log 10 (42  26)  log 10 42  log 10 26
In general,
log 10 ( xy)  log 10 x  log 10 y
The Laws of Logs
Any positive integer could be used as a base instead
of 10, so we get:
log a ( xy)  log a x  log a y
A similar rule holds for dividing.
 x
log a    log a x  log a y
 y
If the base is missed out, you should assume it
could be any base e.g. log 2 might be base 10 or
any other number.
The Laws of Logs
SUMMARY
 The Laws of Logarithms are:
•
1. Multiplication law
log a xy  log a x  log a y
•
2. Division law
x
•
3. Power law
log a
y
 log a x  log a y
log a x
k
 k log a x
 The definition of a logarithm:
a
leads to 4.
x
 b

x  log a b
log a 1  0
6.
log a a
k
5.
k
log a a  1
The Laws of Logs
e.g. 1 Express the following in terms of
log 2, log 3 and log 5
1
(a) log 15 (b) log 16
(c) log  
3
Solution:
(a) log 15  log 3  5  log 3  log 5 ( Law 1 )
4
( Law 3 )
 log 2  4 log 2
1
(c) Either log    log 1  log 3 ( Law 2 )
3
( Law 4 )
 0  log 3
(b)
log 16
  log 3
Or
1
1
log    log 3
 3    log 3 ( Law 3 )
The Laws of Logs
e.g. 2 Express log( a b 2 ) in terms of log a and log b
Solution:
We can’t use the power to the front law directly!
( Why not? )
There is no bracket round the ab, so the square
ONLY refers to the b.
So,
2
log( a b )  log a  log b
2
 log a  2 log b
( Law 1 )
( Law 3 )
The Laws of Logs
e.g. 3 Express each of the following as a single
logarithm in its simplest form:
(a) log 5  log 2  log 3 (b) 2 log 10 4 
log 10 25  1
5 2 

 3 
 10 
 log 

 3 
2 log 10 4  1 log 10 25  1
2
Solution: (a) log 5  log 2  log 3  log 
(b)
1
2
2
1
25 2
 log 10 4  log 10
 log 10 10
 2

2
 16  10 
 4  10 
 log 10 
  log 10 32
  log 10 
1
51 



 25 2 
5
This could be simplified to log 10 2  5 log 10 2
The Laws of Logs
Exercise
1. Express the following in terms of log 2, log 3 and log 5
(a) log 25
Ans:
(b) log 6
(a) 2 log 5
(c) log
(b) log 2  log 3
1
10
(c)  log 2  log 5
2. Express log a b in terms of log a and log b
2
Ans: 2 log a  log b
3. Express the following as a single logarithm in
its simplest form:
(a) log 3  log 2  log 5 (b) 3 log 10 2 
Ans: (a)
log
15
2
(b) log 10
16
5
1
2
log 10 16  1
The Laws of Logs
The Laws of Logs
The following slides contain repeats of
information on earlier slides, shown without
colour, so that they can be printed and
photocopied.
For most purposes the slides can be printed
as “Handouts” with up to 6 slides per sheet.
The Laws of Logs
SUMMARY
 The Laws of Logarithms are:
•
1. Multiplication law
log a xy  log a x  log a y
•
2. Division law
x
•
3. Power law
log a
y
 log a x  log a y
log a x
k
 k log a x
 The definition of a logarithm:
a
leads to 4.
x
 b

x  log a b
log a 1  0
6.
log a a
k
5.
k
log a a  1
The Laws of Logs
e.g. 1 Express the following in terms of
log 2, log 3 and log 5
1
(a) log 15 (b) log 16
(c) log  
3
Solution:
(a) log 15  log 3  5  log 3  log 5 ( Law 1 )
4
( Law 3 )
 log 2  4 log 2
1
(c) Either log    log 1  log 3 ( Law 2 )
3
( Law 4 )
 0  log 3
(b)
log 16
  log 3
Or
1
1
log    log 3
 3    log 3 ( Law 3 )
The Laws of Logs
e.g. 2 Express log( a b 2 ) in terms of log a and log b
Solution:
We can’t use the power to the front law directly!
( Why not? )
There is no bracket round the ab, so the square
ONLY refers to the b.
So,
2
log( a b )  log a  log b
2
 log a  2 log b
( Law 1 )
( Law 3 )
The Laws of Logs
e.g. 3 Express each of the following as a single
logarithm in its simplest form:
(a) log 5  log 2  log 3 (b) 2 log 10 4 
log 10 25  1
5 2 

 3 
 10 
 log 

 3 
2 log 10 4  1 log 10 25  1
2
Solution: (a) log 5  log 2  log 3  log 
(b)
1
2
2
1
25 2
 log 10 4  log 10
 log 10 10
 2

2
 16  10 
 4  10 
 log 10 
  log 10 32
  log 10 
1
51 



 25 2 
5
This could be simplified to log 10 2  5 log 10 2