www.le.ac.uk logarithms department of mathematics university of leicester
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www.le.ac.uk
Logarithms
Department of MathematicsUniversity of Leicester
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Contents
Taking Logs
Introduction
What is a Logarithm?
Properties of Logarithms
Inverse of Log
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Introduction
Logarithms are to do with raising numbers to different powers.
If you write an equation in terms of logarithms it’s like phrasing the equation in a different way.
Next
Taking Logs
IntroWhat is a
Logarithm?Properties of Logarithms
Inverse of Log
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IntroductionWhy do we use logarithms?
Next
• Phrasing the equation in a different way sometimes makes it easier to solve.
• Logarithms have a different kind of scale - the difference between two numbers is a ratio rather than a subtraction. Some relationships are more easy to spot if we’re working with logarithms.
Taking Logs
IntroWhat is a
Logarithm?Properties of Logarithms
Inverse of Log
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What is a logarithm?
Next
• A logarithm, or log, is a function.
• It is written as:
• This is read as “log to the base a of b”
• It means:
• eg. because the power that you have to raise 2 by to get 8 is 3.
balog
What power do you have to raise a by to
get b?38log2
Taking Logs
IntroWhat is a
Logarithm?Properties of Logarithms
Inverse of Log
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2
What is a logarithm?Question
What is ?
3 4
Taking Logs
IntroWhat is a
Logarithm?Properties of Logarithms
Inverse of Log
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What is a logarithm?Question
What is the value of a in this expression: ?4625log a
15 25
Taking Logs
IntroWhat is a
Logarithm?Properties of Logarithms
Inverse of Log
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What is a logarithm?
We said that a Logarithm is a function.It looks like this:
It is the inverse of the exponential function.
Next
Taking Logs
IntroWhat is a
Logarithm?Properties of Logarithms
Inverse of Log
x
y
1
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What is a logarithm?Bases
You can use any base you want in a logarithm.
If we don’t write a base on our logarithm then we assume it is to the base 10.
‘ln’ means ‘natural log’ or ‘log to the base e’. e is a constant number, like π, and we sometimes use this because it has patterns that are seen in nature.
Next
Taking Logs
IntroWhat is a
Logarithm?Properties of Logarithms
Inverse of Log
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Inverse of Log
The inverse of is . ie. the inverse of ‘log to the base a’ is ‘a to the power’
In other words:
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balog ba
baba )(log ba ba log
Taking Logs
IntroWhat is a
Logarithm?Properties of Logarithms
Inverse of Log
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Inverse of Log: Proofs
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This means “What power do I have to raise a by to get ab
? The answer is b.
)(log ba a
Taking Logs
IntroWhat is a
Logarithm?Properties of Logarithms
means “What power do I have to raise a by to get b?” If I then take a to the power of this number, I get b.
ba ba log balog
Inverse of Log
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‘Taking Logs’
‘Taking logs’ of both sides means putting the log function round them.
Take logs of to get
To get back, we do the inverse of log, which is ‘a to the power’: .
This simplifies to , which is what we started with.
7310
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)73(log)10(log aa
)73(log)10(log aa aa
7310
Taking Logs
IntroWhat is a
Logarithm?Properties of Logarithms
Inverse of Log
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• So, we can take logs, but how do we work with them once we’ve got them?
• There are 3 main properties of logarithms:
1.
2.
3.
Properties of Logarithms
)log(loglog abba
Next
abab loglog
b
aba logloglog
Taking Logs
IntroWhat is a
Logarithm?Properties of Logarithms
Inverse of Log
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1. When you add 2 logs, you get the log of their product.
Proof:
Properties of Logarithms
)log(loglog abba
Next
abCab
bBb
aAa
C
B
A
10 so ,log
10 so ,log
10 so ,log
)log(loglog1010
1010
101010
)log()log(log
abba
abba
CBA
CBA
Using the laws of indices
Let: Then:
Taking Logs
IntroWhat is a
Logarithm?Properties of Logarithms
Inverse of Log
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2.
Proof:
Properties of Logarithms
Next
b
aba logloglog
Using the laws of indices
b
aba
b
aba
CBA
CB
A
logloglog
1010
1010
1010
10
logloglog
b
aC
b
abBb
aAa
C
B
A
10 so ,log
10 so ,log
10 so ,log
Then: Let:
Taking Logs
IntroWhat is a
Logarithm?Properties of Logarithms
Inverse of Log
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3.
Proof:
So .
Properties of Logarithms
Next
abab loglog
bb
a
abaabLHS a
cancel 10 base logandpower theto10 because ,
logloglog 10101010
b
a
aRHS ab
b
cancel 10 base logandpower theto10 because ,
log1010
RHSLHS
Note: this is , not . balog balog
Taking Logs
IntroWhat is a
Logarithm?Properties of Logarithms
Inverse of Log
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• Remember the definition of log:
• For any positive number a:
• , because
• , because
• doesn’t exist,because is always > 0.
Properties of Logarithms
Next
Taking Logs
IntroWhat is a
Logarithm?Properties of Logarithms
01log a 10 a
1log aa aa 1
)0something(log a xa
bacb ca log
Inverse of Log
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Example
Solve this:
Take logs:
Use property 3:
Rearrange for x:
Next
xx 311 45
)4log()5log( 311 xx
4log)31(5log)1( xx
4log35log
4log5log
x
Taking Logs
IntroWhat is a
Logarithm?Properties of Logarithms
Inverse of Log
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Question
Write as a single logarithm.
4
127 log)log()log( wvu
4
127log wvu
4
1
27
log
w
vu
4
1
27
log
w
vu
Taking Logs
IntroWhat is a
Logarithm?Properties of Logarithms
Inverse of Log
Click here to see a hint
(Hide Hint)
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Question
Solve
15
105log7log)3log( x
5 35
Taking Logs
IntroWhat is a
Logarithm?Properties of Logarithms
Inverse of Log
Click here to see the solution
(Hide Solution)
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Question
Solve 515 103 x
53log
51 5
3log
51log 5
3
51log
Taking Logs
IntroWhat is a
Logarithm?Properties of Logarithms
Inverse of Log
Click here to see the solution (Hide Solution)
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Logarithms give us another way of writing equations.
We can take logs, take inverse logs, or use the definition.
We can use the three properties of logs to simplify equations.
Conclusion
Next
Taking Logs
IntroWhat is a
Logarithm?Properties of Logarithms
Inverse of Log
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