Department of Mathematics University of Leicester

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Department of Mathematics University of Leicester

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Department of Mathematics University of Leicester

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Presentation transcript:

Department of Mathematics University of Leicester Logarithms Department of Mathematics University of Leicester

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

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. Next

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. Next

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? Next

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

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

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 Next

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. Next

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: Next

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. Next

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. Next

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: Next

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: Next

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: Next

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 . Next

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. Next

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: Next

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)

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

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)

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. Next