Logarithms of Products

The first property we discuss is related to the product rule for exponents:

.m n m nb b b

Lets examine

log3(9 · 27) vs log39 + log327.

Note that

log3(9 · 27) = log3243 = 5 35 = 243

and that

log39 + log327 = 2 + 3 = 5. 32 = 9 and 33 = 27

log3(9 · 27) = log39 + log327.

So

Solution

= loga(42).

Using the product rule for logarithms

Example

that is a single logarithm:

Express as an equivalent expression

loga6 + loga7.

loga6 + loga7 = loga(6 · 7)

Logarithms of Powers

The second basic property is related to the power rule for exponents:

.nm mnb b

Solution

= log4x1/2

Using the power rule for logarithms

a) loga6-3 = –3loga6

Example

expression that is a product:

Use the power rule to write an equivalent

a) loga6–3;

4b) log .x

4b) log x

= ½ log4x Using the power rule

for logarithms

Logarithms of Quotients

The third property that we study is similar to the quotient rule for exponents:

.m

m nn

bb

b

Solution

log3(9/y)

Example

Express as an equivalent expression that is a difference of logarithms:

log3(9/y).

= log39 – log3y. Using the quotient

rule for logarithms

Solution

Example

Express as an equivalent expression that is a single logarithm:

loga6 – loga7.

loga6 – loga7 = loga(6/7) Using the quotient

rule for logarithms “in reverse”

Solution

= log4x3 – log4 yz

Example

using individual logarithms of x, y, and z.

Expand to an equivalent expression

334 7

a) log b) logbx xy

yz z

3

4a) log x

yz

= 3log4x – log4 yz

= 3log4x – (log4 y + log4z)

= 3log4x – log4 y – log4z

Using the Properties Together

1/ 33

7 7 b) log logb b

xy xy

z z

71

log3 b

xy

z

71log log

3 b bxy z

1log log 7log

3 b b bx y z

Solution continued

1 1 7log log log

3 3 3b b bx y z

Solution

Example

using a single logarithm.

Condense to an equivalent expression

1log 2log log

3 b b bx y z

1log 2log log

3 b b bx y z = logbx1/3 – logb y2 + logbz

1/ 3

2log logb b

xz

y

3

2logb

z x

y

Solution

Example Simplify: a) log668 b) log33–3.4

a) log668 = 8

b) log33–3.4 = –3.4

8 is the exponent to which you raise 6 in order to get 68.