UNIT 5 WORKSHEET 7 PROPERTIES OF LOGS

[Pages:2]WORKSHEET 7 Properties of Logarithms

The following properties serve to expand or condense a logarithm or logarithmic expression so it can be worked with.

Properties of logarithms

Example

log

a

mn

=

log a

m

+

log a

n

m

log a

=

log a

m

-

log

a

n

n

log a

mn

=

n log a

m

log4 3x = log4 3 + log4 x

log 2

x +1 5

=

log2

(x

+1) - log2

5

log3 (2x +1)3 = 3log3 (2x +1)

Properties of Natural Logarithms

Example

ln mn = ln m + ln n

m ln = ln m - ln n

n

ln mn = n ln m

ln ( x +1)( x - 5) = ln ( x +1) + ln ( x - 5)

x ln = ln x - ln 2

2 ln 73 = 3ln 7

These properties are used backwards and forwards in order to expand or condense a logarithmic expression. Therefore, these skills are needed in order to solve any equation involving logarithms. Logarithms will also be dealt with in Calculus. If a student has a firm grasp on these three simple properties, it will help greatly in Calculus.

Expanding Logarithmic Expressions

Write each of the following as the sum or difference of logarithms. In other words, expand each logarithmic expression.

1)

log2

3x3 y2

5

z

2) log3 5 3 xy2

3) log 4 ( x +1)3 ( x - 2)2

5 3( x + 2)3

5) log2 x -1

7) loga 12x3 y

9)

log4

(

x

+

3)2 (

x+

x 2

-

6)

4)

log

5

6x2 11y5

z

6)

log12

x-7 x+ 2

8) log3

5x5 y3

32

z

10) log2

5x3 y5z3

Condensing Logarithmic Expressions

Rewrite each of the following logarithmic expressions using a single logarithm. Condense each of the following to a single expression. Do not multiply out complex polynomials. Just

leave something like ( x + 5)3 alone.

11) 3log4 x - 5log4 y + 2 log4 z

12) 2 log x + 1 log y 2

13) 1 log 6 + 1 log x + 2 log y

3

3

3

14)

3 4

log3 16

-

1 3

log3

3

x

-

2 log3

y

15) 3log2 ( x - 4) - 2 log2 ( x + 4) + log2 ( x + 2)

16)

1 3

log2

x

+

2 3

log2

y

-

3

17) log3 ( x + 2) + log3 ( x - 2) - log3 ( x + 4)

18) 3log5 x + 2 log5 y + log5 z + 2

19) 2 log ( x +1) + 1 log ( x - 2) - 1 log ( x + 5)

3

3

3

20) 3log2 3 + 5log2 a + 4 log2 b - 5

Practice Using Properties of Logarithms

Use the following information, to approximate the logarithm to 4 significant digits by using the properties of logarithms.

log a

2

0.3562,

log a

3

0.5646,

and

loga 5 0.8271

21) log 6 a 5

22) log 18 a

23) log 100 a

24) log 30 a 4

27) log a 9

25) log 3 a

28) log 3 15 a

26) log 75 a

29) log 542 a

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