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The laws of logarithms

mc-logs2-2009-1

There are a number of rules which enable us to rewrite expressions involving logarithms in different,

yet equivalent, ways. These rules are known as thelaws of logarithms. You will find that your

lecturers use these laws to present answers in different forms, and so you should make yourself aware

of them and how they are used. The laws apply to logarithms of any base but the same base mustbe 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

106 + log102 = log10(6×2) = log1012

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

logAn=nlogA

So, for example

log

1064= 4log106

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

Third Law

logA-logB= logA B

So, subtractinglogBfromlogAresults inlogAB.

For example, we can write

log e15-loge3 = loge15

3= loge5

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

both sides separately on your calculator. www.mathcentre.ac.uk 1 c?mathcentre 2009

Four other useful results are

log1 = 0,logmm= 1 log

1010n=nlogeen=n

The logarithm of 1 to any base is always 0.

The logarithm of a 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)log108 + log105, (b)logx+ logy, (c)log5x+ log3x, (d)loga+ logb2+ logc3.

2. Use the third law to simplify the following.

(a)log1012-log104, (b)logx-logy, (c)log4x-logx.

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

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

4. Simplify7logx-logx5.

Answers

1. (a)log1040, (b)logxy, (c)log15x2, (d)logab2c3.

2. (a)log103, (b)logx

y, (c)log4.

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

(d)20lnxorlnx20, (e)1000 = 103soln1000 = 3ln10.

4.logx2or equivalently2logx.

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