Transcript Log Properties - Mukwonago Area School District

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 40
40 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 40
The value is
So we rewrite: log 40 into
2.059
log (5)(8) = log 5 + log 8
For example: Use log6 5 = .898 and log6 8 =
1.161 to evaluate log3 40 .
6
6
6
6
We know the values of the yellow
portion so we replace it with
.898 + 1.161
Example 2:
Product Property
If a product is being “logged” we can change it
into a sum.
log5 5x
So we rewrite: log5 5x into
log5 (5)(x) = log5 5 + log5 x
Example 3:
Quotient Property
If a quotient is being “logged” we can change it into
a difference.
𝟓
𝒍𝒐𝒈𝟔
𝟖
For example: Use log 5 = .898 and log
6
1.161 to evaluate
We rewrite as follows:
𝟓
𝒍𝒐𝒈𝟔
𝟖
=log6 5 - log6
8
6
8=
Example 3:
For example: Use log6 5 = .898 and log6 8 =
1.161 to evaluate
𝟓
𝒍𝒐𝒈𝟔
𝟖
𝟓
𝒍𝒐𝒈𝟔
𝟖
The value is
=.898 – 1.161
-0.263
=log6 5 - log6
8
Example 4:
Power Property:
𝒍𝒐𝒈𝟒 𝟒𝟗
The value
is
𝟐
𝒍𝒐𝒈𝟒 𝟕 =2 𝒍𝒐𝒈𝟒 𝟕
2.808
=2(1.404)
Rewrite: Use log4 7 = 1.404 to evaluate
Example 5: Expand
𝟑
𝟓𝒙
𝒍𝒐𝒈𝟔
𝒚
log6 5x3 - log6 y
log6 5+ log6 x3 - log6 y
log6 5 + 3log6 x - log6 y
Example 6: Expand
𝟐
𝒍𝒐𝒈𝟔 𝟒𝒙𝒚
log6 4x + log6 y2
log6 4 + log6 x + log6 y2
log6 4 + log6 x + 2log6 y
Example 6: Condense
2log6 5 + log6 x - 3log6 y
log6 52 + log6 x - log6 y3
log6 25 x - log6 y3
𝟐𝟓𝒙
𝒍𝒐𝒈𝟔 𝟑
𝒚
Example 7: Condense
4ln x – 3ln x
ln x4 – ln x3
𝒙𝟒
ln 𝟑
𝒙
ln x
Change of Base formula
This will let us
use our
calculators!
𝒍𝒐𝒈𝒄 a =
𝒍𝒐𝒈𝒃 𝒂
𝒍𝒐𝒈𝒃 𝒄
Example:
Evaluate:
𝒍𝒐𝒈𝟑 𝟖 = 𝒙
Can’t do it without trial and error
𝒙
𝟑 =𝟖
𝒍𝒐𝒈𝟑 8 =
𝒍𝒐𝒈 𝟖
𝒍𝒐𝒈 𝟑
Example:
Evaluate:
𝒍𝒐𝒈𝟑 𝟖 = 𝒙
Can’t do it without trial and error
𝒙
𝟑 =𝟖
1.89
𝒍𝒐𝒈𝟑 8 =
𝒍𝒐𝒈 𝟖
𝒍𝒐𝒈 𝟑
Example:
Evaluate:
𝒍𝒐𝒈𝟔 𝟒 = 𝒙
.7737
𝒍𝒐𝒈𝟔 4 =
𝒍𝒐𝒈 𝟒
𝒍𝒐𝒈 𝟔
Example:
Evaluate:
𝒍𝒐𝒈𝟑 𝟕 = 𝒙
𝒍𝒐𝒈𝟑 7 =
𝒍𝒐𝒈 𝟕
𝒍𝒐𝒈 𝟑
p. 510 3-6 all, 8, 12,
16-28 evens, 34-38
evens
Graphing Worksheet