1. Department of Mathematics University of Leicester
Logarithms
Department of Mathematics
University of Leicester

2. Properties of Logarithms
Contents
Introduction
What is a Logarithm?
Inverse of Log
Taking Logs
Properties of Logarithms

3. Properties of Logarithms
Intro
What is a Logarithm?
Inverse of Log
Taking Logs
Properties of Logarithms
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.
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4. Introduction Why do we use logarithms?
What is a Logarithm?
Inverse of Log
Taking Logs
Properties of Logarithms
Introduction Why do we use logarithms?
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.
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5. What is a logarithm? A logarithm, or log, is a function.
Intro
What is a Logarithm?
Inverse of Log
Taking Logs
Properties of Logarithms
What is a logarithm?
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.
What power do you have to raise a by to get b?
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6. What is a logarithm? Question
Intro
What is a Logarithm?
Inverse of Log
Taking Logs
Properties of Logarithms
What is a logarithm? Question
What is ? 2 3 4

7. What is a logarithm? Question
Intro
What is a Logarithm?
Inverse of Log
Taking Logs
Properties of Logarithms
What is a logarithm? Question
What is the value of a in this expression: ? 5 15 25

8. Properties of Logarithms
Intro
What is a Logarithm?
Inverse of Log
Taking Logs
Properties of Logarithms
What is a logarithm?
We said that a Logarithm is a function.
It looks like this:
It is the inverse of the exponential function. 1
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9. What is a logarithm? Bases
Intro
What is a Logarithm?
Inverse of Log
Taking Logs
Properties of Logarithms
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.
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10. Properties of Logarithms
Intro
What is a Logarithm?
Inverse of Log
Taking Logs
Properties of Logarithms
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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11. Properties of Logarithms
Intro
What is a Logarithm?
Inverse of Log
Taking Logs
Properties of Logarithms
Inverse of Log: Proofs
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.
This means “What power do I have to raise a by to get ab ? The answer is b.
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12. Properties of Logarithms
Intro
What is a Logarithm?
Inverse of Log
Taking Logs
Properties of Logarithms
‘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.
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13. Properties of Logarithms
Intro
What is a Logarithm?
Inverse of Log
Taking Logs
Properties of Logarithms
Properties of Logarithms
So, we can take logs, but how do we work with them once we’ve got them?
There are 3 main properties of logarithms:
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14. Properties of Logarithms
Intro
What is a Logarithm?
Inverse of Log
Taking Logs
Properties of Logarithms
Properties of Logarithms
When you add 2 logs, you get the log of their product.
Proof:
Using the laws of indices
Let: Then:
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15. Properties of Logarithms
Intro
What is a Logarithm?
Inverse of Log
Taking Logs
Properties of Logarithms
Properties of Logarithms
Proof:
Using the laws of indices
Then:
Let:
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16. Properties of Logarithms
Intro
What is a Logarithm?
Inverse of Log
Taking Logs
Properties of Logarithms
Properties of Logarithms
Proof: So
Note: this is , not
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17. Properties of Logarithms
Intro
What is a Logarithm?
Inverse of Log
Taking Logs
Properties of Logarithms
Properties of Logarithms
Remember the definition of log:
For any positive number a:
, because
doesn’t exist, because is always > 0.
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18. Properties of Logarithms
Intro
What is a Logarithm?
Inverse of Log
Taking Logs
Properties of Logarithms
Example
Solve this:
Take logs:
Use property 3:
Rearrange for x:
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19. Question Write as a single logarithm.
Intro
What is a Logarithm?
Inverse of Log
Taking Logs
Properties of Logarithms
Question
Write as a single logarithm.
Use properties 1 and 2 of logarithms
(click here to go back to this slide)

20. Properties of Logarithms
Intro
What is a Logarithm?
Inverse of Log
Taking Logs
Properties of Logarithms
Question
Solve 5 15 35

21. Properties of Logarithms
Intro
What is a Logarithm?
Inverse of Log
Taking Logs
Properties of Logarithms
Question
Solve
(because ‘log base 10’ and ‘10 to the power’ are inverses)
(using property 3)

22. Properties of Logarithms
Intro
What is a Logarithm?
Inverse of Log
Taking Logs
Properties of Logarithms
Conclusion
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.
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