“Teach A Level Maths” Vol. 1: AS Core Modules 48: More Laws of Logarithms © Christine Crisp

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Log laws for Multiplying and Dividing We’ll develop the laws by writing an example with the numbers in index form.

A log is just an index, so to write this in index form we need the logs from the calculator. and So,

A log is just an index, so to write this in index form we need the logs from the calculator. and So,

A log is just an index, so to write this in index form we need the logs from the calculator. and So,

A log is just an index, so to write this in index form we need the logs from the calculator. and So, In general,

Any positive integer could be used as a base instead of 10, so we get: A similar rule holds for dividing. If the base is missed out, you should assume it could be any base e.g. might be base 10 or any other number.

SUMMARY The Laws of Logarithms are: 1. Multiplication law 2. Division law 3. Power law The definition of a logarithm: leads to 4. 5. 6.

e.g. 1 Express the following in terms of (a) (b) (c) Solution: (a) ( Law 1 ) (b) ( Law 3 ) (c) Either ( Law 2 ) ( Law 4 ) Or ( Law 3 )

e.g. 2 Express in terms of and 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, ( Law 1 ) ( Law 3 )

e.g. 3 Express each of the following as a single logarithm in its simplest form: (b) Solution: (a) (b) This could be simplified to

Exercise 1. Express the following in terms of (a) (b) (c) Ans: (a) (b) (c) 2. Express in terms of and Ans: 3. Express the following as a single logarithm in its simplest form: (a) (b) (a) (b) Ans:

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SUMMARY The Laws of Logarithms are: 1. Multiplication law 2. Division law 3. Power law The definition of a logarithm: leads to 4. 5. 6.

(a) ( Law 1 ) (b) ( Law 3 ) (c) Either ( Law 2 ) ( Law 4 ) Solution: e.g. 1 Express the following in terms of (c) Or

e.g. 2 Express in terms of and 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, ( Law 1 ) ( Law 3 )

(b) e.g. 3 Express each of the following as a single logarithm in its simplest form: (a) Solution: This could be simplified to