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Www.le.ac.uk Logarithms Department of Mathematics University of Leicester.

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1 Logarithms Department of Mathematics University of Leicester

2 Contents Taking LogsIntroductionWhat is a Logarithm? Properties of Logarithms Inverse of Log

3 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 Intro What is a Logarithm? Properties of Logarithms Inverse of Log

4 Introduction Why 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 Intro What is a Logarithm? Properties of Logarithms Inverse of Log

5 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. What power do you have to raise a by to get b? Taking Logs Intro What is a Logarithm? Properties of Logarithms Inverse of Log

6 2 What is a logarithm? Question What is ? 34 Taking Logs Intro What is a Logarithm? Properties of Logarithms Inverse of Log

7 5 What is a logarithm? Question What is the value of a in this expression: ? 1525 Taking Logs Intro What is a Logarithm? Properties of Logarithms Inverse of Log

8 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 Intro What is a Logarithm? Properties of Logarithms Inverse of Log 1

9 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 Intro What is a Logarithm? Properties of Logarithms Inverse of Log

10 Inverse of Log The inverse of is. ie. the inverse of ‘log to the base a’ is ‘a to the power’ In other words: Next Taking Logs Intro What is a Logarithm? Properties of Logarithms Inverse of Log

11 Inverse of Log: Proofs Next This means “What power do I have to raise a by to get a b ? The answer is b. Taking Logs Intro What 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. Inverse of Log

12 ‘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. Next Taking Logs Intro What is a Logarithm? Properties of Logarithms Inverse of Log

13 So, we can take logs, but how do we work with them once we’ve got them? There are 3 main properties of logarithms: Properties of Logarithms Next Taking Logs Intro What is a Logarithm? Properties of Logarithms Inverse of Log

14 1. When you add 2 logs, you get the log of their product. Proof: Properties of Logarithms Next Using the laws of indices Let: Then: Taking Logs Intro What is a Logarithm? Properties of Logarithms Inverse of Log

15 2. Proof: Properties of Logarithms Next Using the laws of indices Then: Let: Taking Logs Intro What is a Logarithm? Properties of Logarithms Inverse of Log

16 3. Proof: So. Properties of Logarithms Next Note: this is, not. Taking Logs Intro What is a Logarithm? Properties of Logarithms Inverse of Log

17 Remember the definition of log: For any positive number a:, because doesn’t exist, because is always > 0. Properties of Logarithms Next Taking Logs Intro What is a Logarithm? Properties of Logarithms Inverse of Log

18 Example Solve this: Take logs: Use property 3: Rearrange for x: Next Taking Logs Intro What is a Logarithm? Properties of Logarithms Inverse of Log

19 Use properties 1 and 2 of logarithms (click here to go back to this slide) Question Write as a single logarithm. Taking Logs Intro What is a Logarithm? Properties of Logarithms Inverse of Log

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

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

22 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 Intro What is a Logarithm? Properties of Logarithms Inverse of Log

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