More Logarithms and Indices Lucan Community College Leaving Certificate Mathematics Higher Level Mr Duffy & Ms McKelvey More Logarithms and Indices © Ciarán Duffy

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