log properties. because logs are really exponents they have similar properties to exponents. recall...

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