The laws of logarithms









hp calculators

logarithm of a given number is the exponent that a base number must have to equal formula is very useful to change logarithms from one base to another:.
s logarithms


hp calculators

Common logarithms are also called “log to base 10” and the common logarithm of a number “x” is written. LOG10 x or just LOG x. Natural logarithms are also 
sLog


MATHEMATICS 0110A CHANGE OF BASE Suppose that we have

(The numbers the calculator was displaying immediately after hitting LN or LOG were quite different in these 2 calculations but the final answer is the same.) 
Change of Base


1.2 The logarithm is the index.

6 sept. 2007 Now suppose that we calculate the logarithms of each of these ... deed there are only three different bases one encounters at all often in.





6.2 Properties of Logarithms

We apply the Change of Base formula with a = 3 and b = 10 to obtain 32 = 102 log(3). Typing the latter in the calculator produces an answer of 9 as required. 2.
S&Z . & .


What is a logarithm? Log base 10

Graphing with logarithms. Another powerful use of logarithms comes in graphing. For example exponential functions are tricky to compare visually.
logarithms


The laws of logarithms

logarithms to be rewritten in a variety of different ways. The laws apply to logarithms of any base but the same base must be used throughout a calculation.
mc bus loglaws


2.19 What is a logarithm ?

We use logarithms to write expressions involving powers in a different form. Your calculator will be able to calculate logarithms to bases 10 and e.





Logarithms - changing the base

This leaflet gives this formula and shows how to use it. A formula for change of base. Suppose we want to calculate a logarithm to base 2. The formula states.
mc logs


Chapter 4: Exponential and Logarithmic Functions

Since exponential functions have different bases we will define corresponding logarithms of different bases as well. Logarithm. The logarithm (base b) function 
Chapter


217631 The laws of logarithms

The laws of logarithms

mc-bus-loglaws-2009-1

Introduction

There are a number of rules known as thelaws of logarithms. These allow expressions involving logarithms to be rewritten in a variety of different ways. Thelaws apply to logarithms of any base but the same base must be used throughout a calculation.

The laws of logarithms

The three main laws are stated here:

First Law

logA+ logB= logAB This law tells us how to add two logarithms together. AddinglogAandlogBresults in the logarithm of the product ofAandB, that islogAB.

For example, we can write

log

105 + log104 = log10(5×4) = log1020

The same base, in this case 10, is used throughout the calculation. You should verify this by evaluating both sides separately on your calculator.

Second Law

logA-logB= logA B

So, subtractinglogBfromlogAresults inlogAB.

For example, we can write

log e12-loge2 = loge12

2= loge6

The same base, in this case e, is used throughout the calculation. You should verify this by evaluating

both sides separately on your calculator.

Third Law

logAn=nlogA

So, for example

log

1053= 3log105

You should verify this by evaluating both sides separately on your calculator.

Two other important results are

www.mathcentre.ac.uk 1 c?mathcentre 2009 log1 = 0,logmm= 1 The logarithm of 1 to any base is always 0, and the logarithm ofa number to the same base is always 1. In particular, log

1010 = 1,andlogee = 1

Exercises

1. Use the first law to simplify the following.

a)log106 + log103, b)logx+ logy, c)log4x+ logx, d)loga+ logb2+ logc3.

2. Use the second law to simplify the following.

a)log106-log103, b)logx-logy, c)log4x-logx.

3. Use the third law to write each of the following in an alternative form.

a)3log105, b)2logx, c)log(4x)2, d)5lnx4, e)ln1000.

4. Simplify3logx-logx2.

Answers

1. a)log1018, b)logxy, c)log4x2, d)logab2c3.

2. a)log102, b)logx

y, c)log4.

3. a)log1053orlog10125, b)logx2, c)2log(4x), d)20lnxorlnx20,

e)1000 = 103soln1000 = 3ln10.

4.logx.

www.mathcentre.ac.uk 2 c?mathcentre 2009

The laws of logarithms

mc-bus-loglaws-2009-1

Introduction

There are a number of rules known as thelaws of logarithms. These allow expressions involving logarithms to be rewritten in a variety of different ways. Thelaws apply to logarithms of any base but the same base must be used throughout a calculation.

The laws of logarithms

The three main laws are stated here:

First Law

logA+ logB= logAB This law tells us how to add two logarithms together. AddinglogAandlogBresults in the logarithm of the product ofAandB, that islogAB.

For example, we can write

log

105 + log104 = log10(5×4) = log1020

The same base, in this case 10, is used throughout the calculation. You should verify this by evaluating both sides separately on your calculator.

Second Law

logA-logB= logA B

So, subtractinglogBfromlogAresults inlogAB.

For example, we can write

log e12-loge2 = loge12

2= loge6

The same base, in this case e, is used throughout the calculation. You should verify this by evaluating

both sides separately on your calculator.

Third Law

logAn=nlogA

So, for example

log

1053= 3log105

You should verify this by evaluating both sides separately on your calculator.

Two other important results are

www.mathcentre.ac.uk 1 c?mathcentre 2009 log1 = 0,logmm= 1 The logarithm of 1 to any base is always 0, and the logarithm ofa number to the same base is always 1. In particular, log

1010 = 1,andlogee = 1

Exercises

1. Use the first law to simplify the following.

a)log106 + log103, b)logx+ logy, c)log4x+ logx, d)loga+ logb2+ logc3.

2. Use the second law to simplify the following.

a)log106-log103, b)logx-logy, c)log4x-logx.

3. Use the third law to write each of the following in an alternative form.

a)3log105, b)2logx, c)log(4x)2, d)5lnx4, e)ln1000.

4. Simplify3logx-logx2.

Answers

1. a)log1018, b)logxy, c)log4x2, d)logab2c3.

2. a)log102, b)logx

y, c)log4.

3. a)log1053orlog10125, b)logx2, c)2log(4x), d)20lnxorlnx20,

e)1000 = 103soln1000 = 3ln10.

4.logx.

www.mathcentre.ac.uk 2 c?mathcentre 2009
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