WHAT ARE THE PROPERTIES OF LOGARITHMS?

Evaluate the following:

4

64log

64log

16log

4log

2log

2

2

2

2

2

Recall: a logarithm is an exponent. So in each case, we are looking for the exponent of 2 to get a number. In the first example, the answer is 1

1

Because 221 246

4

4

64log

64log

16log

4log

2log

2

2

2

2

2 1

246

4

Compare example 4 with examples 2 and 3What do you notice?

If we multiply 4 and 16 we’ll get 64

If we add 2 and 4 we get 6

There is a relationship between these examples.

16log4log64log 222

Another way of seeing this:

164log64log 22

16log4log164log 222

But we know that

416log

24log

2

2

So we can

substitute to get:

42So we found that:

6164log64log 22 We should check our answer:

664log2 6426 This is true, so our answer is correct.

In the same manner, we can create rules for the logarithm of a quotient and the logarithm of something raised to a powerIf we do that, we have the following properties of logarithms1) The product rule:

caca bbb logloglog 2) The quotient rule:

cac

abbb logloglog

3) The power rule:

aca bc

b loglog

Example – Use the laws of logarithms to expand the logarithm

32log yxbThere are two laws we can apply here: the power and the product rule. We must do the product rule first.

32 loglog yx bb Now we can apply the power rule:

yx bb log3log2

Example – Use the laws of logarithms to expand the logarithm

y

xblog

Quotient rule first

yx bb loglog We can apply the power rule also, if we change the form of the square root of x

yx bb loglog 2

1

yx bb loglog2

1

Example – Use the laws of logarithms to write the following expression as a single logarithm

yx log3log2 Before we use quotient rule, we must use the power rule32 loglog yx Now the quotient rule

3

2

logy

x

Example – Use the laws of logarithms to write the following expression as a single logarithm

znm loglog3log2

1

znm logloglog 32

1

znm logloglog 3 zmn loglog 3

z

mn3log