Properties of Logarithms

Since logs and exponentials of the same base are inverse functions of each other they “undo” each other.

xxfaxf ax log1

Remember that: xffxff 11 and

This means that: log1 xaff xa

xaff xa log1

5log22

inverses “undo” each each other

= 57

3 3log = 7

1.

2.

3.

CONDENSED EXPANDED

Properties of Logarithms

NM aa loglog

=

=

= NM aa loglog

= Mr alog

(these properties are based on rules of exponents since logs = exponents)

MNalog

N

Malog

ra Mlog

Using the log properties, write the expression as a sum and/or difference of logs (expand).

3 2

4

6logc

ab

using the second property: 3

2

64

6 loglog cab

When working with logs, re-write any radicals as rational exponents.

3

2

4

6log

c

ab

using the first property: 3

2

64

66 logloglog cba

using the third property: cba 666 log3

2log4log

NMN

Maaa logloglog

NMMN aaa logloglog

MrM ar

a loglog

Using the log properties, write the expression as a single logarithm (condense).

yx 33 log2

1log2

using the third property:

using the second property:

2

1

2

3log

y

x

2

1

32

3 loglog yx MrM a

ra loglog

NMN

Maaa logloglog

this direction

this direction

More Properties of Logarithms

NMNM aa loglog then , If

NMNM aa then ,loglog If

This one says if you have an equation, you can take the log of both sides and the equality still holds.

This one says if you have an equation and each side has a log of the same base, you know the "stuff" you are taking the logs of are equal.

8log2

(2 to the what is 8?)

3

16log2

(2 to the what is 16?)

4

10log2

(2 to the what is 10?)

There is an answer to this and it must be more than 3 but less than 4, but we can't do this one in our head.

x10log2

Let's put it equal to x and we'll solve for x.

Change to exponential form.

102 x

NMNM aa loglog then , If

use log property & take log of both sides (we'll use common log)

10log2log xuse 3rd log property

MrM ar

a loglog

10log2log xsolve for x by dividing by log 2

2log

10logx 32.3use calculator to

approximate

32.3

Check by putting 23.32 in your calculator (we rounded so it won't be exact)

Change-of-Base Formula

The base you change to can be any base so generally we’ll want to change to a base so we can use our calculator. That would be either base 10 or base e.

LOG

“common” log base 10

LN

“natural” log base e

a

M

log

log

a

M

ln

ln

Example for TI-83

If we generalize the process we just did we come up with the:

a

MM

b

ba log

loglog

Use the Change-of-Base Formula and a calculator to approximate the logarithm. Round your answer to three decimal places.

16log3

Since 32 = 9 and 33 = 27, our answer of what exponent to put on 3 to get it to equal 16 will be something between 2 and 3.

3ln

16ln16log3

put in calculator

524.2

Acknowledgement

I wish to thank Shawna Haider from Salt Lake Community College, Utah USA for her hard work in creating this PowerPoint.

www.slcc.edu

Shawna has kindly given permission for this resource to be downloaded from www.mathxtc.com and for it to be modified to suit the Western Australian Mathematics Curriculum.

Stephen CorcoranHead of MathematicsSt Stephen’s School – Carramarwww.ststephens.wa.edu.au