Log Properties

Because logs are REALLY exponents they have similar properties to exponents.

Recall that when we MULTIPLY like bases we ADD the exponents. (Simplify (32 )(310 )

And when we DIVIDE like bases we SUBTRACT the exponents. (Simplify (32 )(310 )

Something similar happens with logs…. (And of course, whatever holds for logs also holds for ln.

Example 1:Product Property

If a product is being “logged” we can change it into a sum.

log3 4040 is a can be a lot of different products. For

example: 4 and 10 or 8 and 5. They tell you what to factor it into.

Example 1:Product Power

log6 40For example: Use log6 5 = .898 and log6 8 =

1.161 to evaluate .log3 40So we rewrite: log6 40 into log6 (5)(8) = log6 5 + log6 8

We know the values of the yellow portion so we replace it with

.898 + 1.161

The value is 2.059

Example 2:Product PropertyIf a product is being “logged” we can change it

into a sum.

log5 5xSo we rewrite: log5 5x into log5 (5)(x) = log5 5 + log5 x

Example 3:Quotient PropertyIf a quotient is being “logged” we can change it into

a difference.

𝒍𝒐𝒈𝟔𝟓𝟖

For example: Use log6 5 = .898 and log6 8 = 1.161 to evaluate

We rewrite as follows:

=log6 5 - log6 8

Example 3:For example: Use log6 5 = .898 and log6 8 =

1.161 to evaluate

=log6 5 - log6 8

=.898 – 1.161

The value is -0.263

Example 4:Power Property:

𝒍𝒐𝒈𝟒𝟒𝟗Rewrite: Use log4 7 = 1.404 to evaluate

=2(1.404)

=2 The value is

2.808

Example 5: Expand

𝒍𝒐𝒈𝟔𝟓𝒙𝟑

𝒚log6 5x3 - log6 y

log6 5+ log6 x3 - log6 y

log6 5 + 3log6 x - log6 y

Example 6: Expand

𝒍𝒐𝒈𝟔𝟒 𝒙 𝒚𝟐

log6 4x + log6 y2

log6 4 + log6 x + log6 y2

log6 4 + log6 x + 2log6 y

Example 6: Condense2log6 5 + log6 x - 3log6 y

log6 52 + log6 x - log6 y3

log6 25 x - log6 y3

Example 7: Condense4ln x – 3ln x

ln x4 – ln x3

lnln x

Change of Base formulaThis will let us

use our calculators!

a =

Example: Evaluate:

Can’t do it without trial and error

8 =

Example: Evaluate:

Can’t do it without trial and error

8 = 1.89

Example: Evaluate:

4 =

.7737

Example: Evaluate:

7 =

p. 510 3-6 all, 8, 12, 16-28 evens, 34-38

evensGraphing Worksheet