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