47: More Logarithms and Indices “Teach A Level Maths” Vol. 1: AS Core Modules 47: More Logarithms and Indices © Christine Crisp

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We need to be able to change between index forms for numbers and log forms. We use We’ll also develop some more laws of logs.

e.g. Write the following in a form using logarithms: (b) Solution: The index, 3, is the log of 64 and 4 is the base. (b) e.g. Write the following without using logarithms: (a) (b) Solution: (a) (b)

Exercises 1. Write the following in a form using logarithms: (a) (b) 2. Write the following without using logarithms: (a) (b) Solution: 1(a) (b) 2(a) (b)

We are not solving an equation! Simplifying Logs Some logs can be simplified. We are not solving an equation! e.g. 1 Simplify This log can be simplified because we can write 9 in index form using the base 3. The base, 3, is now the same as the base of the log So, since a log is an index! In general,

Simplifying Logs e.g. 2. Simplify (a) (b) Solution: (a) (b)

Exercises 1. Simplify the following log expressions: (a) (b) (c) (d) Solution (a) (b) (c) (d)

2 useful results There are 2 special cases we can get directly from the definition of a log. Let x = 0, By the law of indices, So, for all values of the base

2 useful results There are 2 special cases we can get directly from the definition of a log. Let b = a, Then x = 1

2 useful results There are 2 special cases we can get directly from the definition of a log. Let b = a, Then x = 1 So,

SUMMARY The Definition of a Logarithm Three Laws of Logarithms

Exercises 1. Simplify the following: (a) (b) (c) (d) (a) 1 Ans: (b) 0 (c) 19 (d) b

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e.g. Write the following in a form using logarithms: (b) e.g. Write the following without using logarithms: Solution: (a) Solution:

Simplifying Logs e.g. 2. Simplify (a) (b) (b) Solution: (a)

There are 2 special cases we can get directly from the definition of a log. Let x = 0, 2 useful results So, By the law of indices, for all values of the base

Let b = a, x = 1 Then So,

Three Laws of Logarithms The Definition of a Logarithm SUMMARY