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